Sufficient Conditions of 6-Cycles Make Planar Graphs DP-4-Colorable

Abstract

In simple graphs, DP-coloring is a generalization of list coloring and thus many results of DP-coloring generalize those of list coloring. Xu and Wu proved that every planar graph without 5-cycles adjacent simultaneously to 3-cycles and 4-cycles is 4-choosable. Later, Sittitrai and Nakprasit showed that if a planar graph has no pairwise adjacent 3-, 4-, and 5-cycles, then it is DP-4-colorable, which is a generalization of the result of Xu and Wu. In this paper, we extend the results on 3-, 4-, 5-, and 6-cycles by showing that every planar graph without 6-cycles simultaneously adjacent to 3-cycles, 4-cycles, and 5-cycles is DP-4-colorable, which is also a generalization of previous studies as follows: every planar graph G is DP-4-colorable if G has no 6-cycles adjacent to i-cycles where $i\in \{3,4,5\}$.

1. Introduction

The concept of list coloring was introduced independently by Vizing [1] and Erdős et al. [2]. A k-list assignment L of a graph G assigns for each vertex v in G a list $L\left(v\right)$ of k colors. An L-coloring is a proper coloring c such that $c\left(v\right)\in L\left(v\right)$ for each v in $V\left(G\right)$. A graph G is L-colorable if G has an L-coloring. If G is L-colorable for any k-list assignment L, then G is said to be k-choosable.

DP-coloring is a generalization of list coloring. Dvořák and Postle [3] introduced the concept of DP-coloring and they called it correspondence coloring. Later on, it is called DP-coloring by Bernshteyn et al. [4].

Assume L is an assignment of a graph G. H is a cover of G if it admits all the following properties:

(i)

Its vertex set $V\left(H\right)$ is ${\bigcup }_{v\in V\left(G\right)}(\left\{v\right\}\times L\left(v\right))=\{(v,c):v\in V\left(G\right),c\in L\left(v\right)\};$

(ii)

$H\left[\right\{v\}\times L(v\left)\right]$ is a complete graph for every $v\in V\left(G\right);$

(iii)

The set $E_H(\left\{u\right\}\times L\left(u\right),\left\{v\right\}\times L\left(v\right))$ is a matching (empty matching is allowable) for each $uv\in E\left(G\right)$.

(iv)

If $uv\notin E\left(G\right),$ then there are no edges of H connect $\left\{u\right\}\times L\left(u\right)$ and $\left\{v\right\}\times L\left(v\right)$.

An independent set in a cover H of a graph G with size $\left|V\right(G\left)\right|$ is called an $(H,L)$-coloring of G. If every cover H with any k-assignment L of a graph G admits an $(H,L)$-coloring for $G,$ then we say that G is DP-k-colorable. The minimum k in which a graph G is DP-k-colorable is called the DP-chromatic number of G and denoted by ${\chi }_{DP}\left(G\right)$.

If edges on H are defined to match exactly identical colors between $L\left(u\right)$ and $L\left(v\right)$ for each $uv\in E\left(G\right),$ then G admits an $(H,L)$-coloring is equivalent to G is L-colorable. Consequently, DP-coloring is a generalization of list coloring. Furthermore, this implies that ${\chi }_{DP}\left(G\right)\ge {\chi }_l\left(G\right).$

Dvořák and Postle [3] proved that for every planar graph G, ${\chi }_{DP}\left(G\right)\le 5$, which extends a seminal result by Thomassen [5] on list coloring. Meanwhile, Voigt [6] constructed an example of a non-4-choosable planar graph (and thus, not DP-4-colorable). It motivates the investigation to obtain sufficient conditions for being DP-4-colorable of planar graphs. Kim and Ozeki [7] proved that every planar graph is DP-4-colorable if it does not contain k-cycles for each $k=3,4,5,6$. Kim et al. [8] proved that every planar graph is DP-4-colorable if it contains neither 7-cycles nor butterflies. In [9], Kim and Yu proved that every planar graph is DP-4-colorable if it does not contain triangles adjacent to 4-cycles, which extends the result on 3- and 4-cycles. In 2019, Liu and Li [10] improved the previous result of Kim and Yu [9] by relaxing the condition of one triangle into two triangles. Chen et al. [11] showed that every planar graph that contains no 4-cycles adjacent to k-cycles where $k=5,6$ is DP-4-colorable. Liu et al. [12] extended the result of Kim and Ozeki [7] on 3-, 5-, and 6-cycles by proving that every planar graph contains no k-cycles adjacent to triangles is DP-4-colorable. Xu and Wu [13] proved that every planar graph, which contains no 5-cycles adjacent simultaneously to 3-cycles and 4-cycles is 4-choosable. Recently, Sittitrai and Nakprasit [14] showed that every planar graph that contains no pairwise adjacent 3-, 4-, and 5-cycle is DP-4-colorable which generalizes the result of Xu and Wu [13].

In this work, the results on 3-, 4-, 5-, and 6-cycles are extended by the result on Theorem 1, which generalizes the aforementioned results by Chen et al. [11] and Liu et al. [12].

Theorem 1.

Every planar graph without 6-cycles simultaneously adjacent to 3-cycles, 4-cycles, and 5-cycles is DP-4-colorable.

Then we have the following two Corollaries. Moreover, some results on [11,12] are some part of Corollary 1 for $i=4$ and $i=3$, respectively.

Corollary 1.

Every planar graph without 6-cycles adjacent to i-cycles is DP-4-colorable for each $i\in \{3,4,5\}$.

Corollary 2.

Every planar graph without 6-cycles simultaneously adjacent to i-cycles and j-cycles is DP-4-colorable for each $i,j\in \{3,4,5\}$ and $i\ne j$.

2. Preliminaries

First, some notations and definitions are introduced in this section. Let G be a plane graph. The vertex set, edge set, and face set of the graph G are denoted, respectively, by $V\left(G\right),E\left(G\right),$ and $F\left(G\right).$ We use $B\left(f\right)$ to denote the boundary of a face $f.$ Two faces f and g are adjacent if $B\left(f\right)$ and $B\left(g\right)$ are adjacent. A wheel $W_n$ is a graph of n vertices formed by connecting all vertices of an $(n-1)$-cycle (these vertices are called external vertices) to a single vertex (hub). A k-vertex, $k^{+}$-vertex, and $k^{-}$-vertex is a vertex of degree k, at least k, and at most $k,$ respectively. Similar notation is applied to cycles and faces.

Note that some faces may appear several times in the order. If a face is incident to at least two $5^{+}$-vertices (respectively, exactly one $5^{+}$-vertex, no $5^{+}$-vertices), it is called rich (semi-rich, poor, respectively).

A semi-rich 5-face is a proper semi-rich 5-face if each incident edge with two endpoints of degree 4 is on the boundary of a 3-face, otherwise it is called an improper semi-rich 5-face.

A bounded face is an extreme face if it has a vertex incident to the unbounded face. An inner face is a bounded face but is not an extreme face.

An edge $uv$ is a chord in an embedding cycle C if $u,v\in V\left(C\right)$ but $uv$ is not in $E\left(C\right).$ If a chord is inside $C,$ then it is called an internal chord, otherwise it is called an external chord. A graph $C(m,n)$ is obtained from a cycle $x_1{x}_2\dots x_{m+n-2}$ with an internal chord $x_1{x}_m.$ For example, cycles $uvw$ and $vwxyz$ form $C(3,5).$ A graph $C(l,m,n)$ is obtained from a cycle $x_1{x}_2\dots x_{l+m+n-4}$ with internal chords $x_1{x}_l$ and $x_1{x}_{l+m-2}.$ The previous definition can be extended similarly to a graph $C(m,n,p,q).$ The graphs $int\left(C\right)$ and $ext\left(C\right)$ are induced by vertices inside and outside a cycle $C,$ respectively. A separating cycle C is a cycle with non-empty $int\left(C\right)$ and $ext\left(C\right).$

Let $\mathcal{A}$ denote the family of planar graphs without $6-$cycle simultaneously adjacent 3-, 4-, and 5-cycle.

To prove that every planar graph without 6-cycles simultaneously adjacent to 3-cycles, 4-cycles, and 5-cycles is DP-4-colorable, we prove a stronger result as follows.

Theorem 2.

If $G\in \mathcal{A}$ with a precolored 3-cycle, then the precoloring can be extended to be a DP-4-coloring of G.

3. Structures

Let G be a minimal counterexample to Theorem 2 with respect to the order $\left|V\right(G\left)\right|.$ Then, (i) $G\in \mathcal{A}$ and (ii) G is a minimal graph with a precoloring of a 3-cycle that cannot be extended to be a DP-4-coloring in $G.$ Some tools in [14] are used to deal with graphs satisfying (ii). We assume that G contains a 3-cycle since every planar graph without 3-cycles is DP-4-colorable [9].

Thus we let $C_0$ be a 3-cycle in G that is precolored.

Lemma 1

(Lemma 3.1 in [14]). G has no separating 3-cycles (See the proof in Lemma A1).

It follows from Lemma 1 that we may assume $C_0$ to be the boundary of the unbounded face of $G.$

Lemma 2

(Lemma 3.3 in [14]). Each vertex in $int\left(C_0\right)$ has degree at least four (See the proof in Lemma A3).

Lemma 3.

The following statements hold.

(i)

A bounded $6^{-}$-face has its boundary as a cycle.

(ii)

If a bounded $k_1$-face f and a bounded $k_2$-face g with $k_1+k_2\le 8$ are adjacent, then $B\left(f\right)\cup B\left(g\right)=C(k_1,k_2).$

(iii)

Let a bounded 3-face f and a bounded 4-face g be adjacent. If f or g is adjacent to a bounded 3-face h, then $B\left(f\right)\cup B\left(g\right)\cup B\left(h\right)$ is a 6-cycle with two internal chords.

Proof. 

(i)

Clearly, a boundary of a $5^{-}$-face is a cycle. Consider a bounded 6-face $f.$ A boundary closed walk is in a form of $uvwxywu$ if $B\left(f\right)$ is not a cycle. By Lemma 2, u or x has degree at least $4.$ It follows that $uvw$ or $xyw$ is a separating 3-cycle, contrary to Lemma 1.

(ii)

It suffices to show that $B\left(f\right)$ and $B\left(g\right)$ share exactly two vertices.

If $B\left(f\right)=uvw,B\left(g\right)=vwx$ and $u=x,$ then f or g is the unbounded face, a contradiction.

If $B\left(f\right)=uvw,B\left(g\right)=vwxy$ and $u=x$ or $y,$ then $d\left(w\right)=2$ or $d\left(v\right)=2,$ which contradicts Lemma 2.

If $B\left(f\right)=uvw,B\left(g\right)=vwxyz$ and $u=x$ or $z,$ then $d\left(w\right)=2$ or $d\left(v\right)=2,$ which contradicts Lemma 2. If $B\left(f\right)=uvw,B\left(g\right)=vwxyz$ and $u=y,$ then $vyz$ or $wxy$ is a separating 3-cycle, which contradicts Lemma 1.

If $B\left(f\right)=stuv,B\left(g\right)=uvwx$ and $s=w,$ then $d\left(v\right)=2,$ which contradicts Lemma 2. If $B\left(f\right)=stuv,B\left(g\right)=uvwx$ and $s=x,$ then $utx$ or $vwx$ is a separating 3-cycle, which contradicts Lemma 1. The remaining cases are similar.

(iii)

Lemma 3 (ii) yields that $B\left(f\right)\cup B\left(g\right)$ is a 5-cycle with one chord. Similar to the proof of Lemma 3 (ii), one can show that $B\left(h\right)$ and $B\left(f\right)\cup B\left(g\right)$ share exactly two vertices. This yields a desired result.

□

Lemma 4.

If C is a 6-cycle and has a triangular chord, then C has only one chord. Moreover, every 6-cycle has at most one internal chord.

Proof. 

Let C be a 6-cycle $tuvxyz$ and let $tv$ be its triangular chord. Suppose to the contrary that C has at least two chords. Since C is adjacent to a 3-cycle $tuv$ and a 5-cycle $uvxyz$, it suffices to show that C is adjacent to a 4-cycle. By symmetry, we assume another chord e of C is $ux,uy,tx,ty,$ or $xz.$

If $e=ux,$ then C is adjacent to a 4-cycle $tuxv.$

If $e=uy,$ then C is adjacent to a 4-cycle $uvxy.$

If $e=tx,$ then C is adjacent to a 4-cycle $tuvx.$

If $e=ty,$ then C is adjacent to a 4-cycle $vxyt.$

If $e=xz,$ then C is adjacent to a 4-cycle $tvxz.$

Thus, C has exactly one chord. Note that C has a triangular chord if C has at least two internal chords. It follows that every 6-cycle has at most one internal chord. □

A cluster in a plane graph G is a subgraph of G consisting of 3-cycles from a minimal set of bounded 3-faces such that they are not adjacent to other bounded 3-faces outside the set. A k-cluster is formed by k bounded 3-faces. An adjacent face of an i-cluster $H_i$ is a face that is adjacent to some bounded 3-face in $H_i$. Since $G\in \mathcal{A}$, one can observe that every cluster in G is a $4^{-}$-cluster where a 4-cluster is isomorphic to $W_5$.

Lemma 5.

The following statements hold.

(i)

If a 4-face f is adjacent to an inner 3-face g, then f is not adjacent to other inner 3-faces and f is not adjacent to any 4-faces.

(ii)

If an inner 3-face f is adjacent to a 5-face $g,$ then f and g are not adjacent to any 4-faces.

(iii)

Every adjacent face of a 2-cluster is a $6^{+}$-face or the unbounded 3-face D.

(iv)

Every adjacent face of a $3^{+}$-cluster is a $7^{+}$-face or the unbounded 3-face D.

Proof. 

(i)

Let f be a 4-face adjacent to an inner 3-face g and another face h.

Suppose to the contrary that h is an inner 3-face or a 4-face.

If h is an inner 3-face, then $B\left(f\right)\cup B\left(g\right)\cup B\left(h\right)$ is a 6-cycle with two internal chords by Lemma 3 (iii), contrary to Lemma 4.

If h is a 4-face, then Lemma 3 (ii) yields a 6-cycle from $B\left(f\right)\cup B\left(h\right)$, which is adjacent to a 5-cycle from $B\left(f\right)\cup B\left(g\right)$, a 4-cycle from $B\left(f\right)$, and a 3-cycle from $B\left(g\right)$, contrary to $G\in \mathcal{A}.$

(ii)

Let an inner 3-face f and a 5-face g be adjacent. Lemma 3 (ii) yields that $B\left(f\right)\cup B\left(g\right)$ contains a 6-cycle. Thus, f or g is not adjacent to any 4-faces since $G\in \mathcal{A}$.

(iii)

Let f and g be bounded 3-faces in a 2-cluster $H_2$ and let h be a bounded face adjacent to f. By the definition, h is not a bounded 3-face.

If h is a 4-face, then Lemma 3 (iii) yields that $B\left(f\right)\cup B\left(h\right)\cup B\left(g\right)$ contains a 6-cycle with two internal chords, contrary to Lemma 4.

If h is a 5-face, then it follows from Lemmas 3 (i) and (ii) that a 6-cycle from $B\left(f\right)\cup B\left(h\right)$ is adjacent to a 5-cycle from $B\left(h\right)$, a 4-cycle from $B\left(f\right)\cup B\left(g\right)$, and 3-cycle from $B\left(f\right)$, contrary to $G\in \mathcal{A}.$

Thus, h is a $6^{+}$-face or the unbounded face.

(iv)

Let $f_1,f_2,$ and $f_3$ be the bounded 3-faces of $3^{+}$-cluster $H_3$ in a consecutive order.

By similar arguments as in the proof of (iii), it follows that $H_3$ cannot be adjacent to a bounded $5^{-}$-face.

Let $H_3$ be adjacent to a 6-face $f_4$. By Lemma 3 (ii) and an argument similar to its proof, one can show that $H_3$ is a 5-cycle with two chords. Since $B\left(f_4\right)$ is a 6-cycle by Lemma 3 (i), we have a 6-cycle adjacent to a 3-, a 4-, and a 5-cycle in $H_3$, contrary to $G\in \mathcal{A}.$

If $H_3$ is adjacent to a 6-face $f_4$, then by Lemma 3 (ii), a 6-cycle $B\left(f_4\right)$ is adjacent to a 3-, a 4-, and a 5-cycle, which are in $H_3$, contrary to $G\in \mathcal{A}$.

□

For Corollary 3 (i), it is proved by the fact that every $5^{+}$-vertex is not adjacent to four consecutive bounded 3-faces. Thus, each $5^{+}$-vertex has at least two $4^{+}$-faces. For Corollary 3 (ii), it is proved by Lemmas 5 (iii) and (iv) that each 3-face in $H_2^{+}$ is not adjacent to a 5-face. Thus, each $5^{+}$-vertex has at least three $4^{+}$-faces.

Corollary 3.

Let v be a k-vertex in G where $v\notin V\left(C_0\right)$ and $k\ge 5.$ It follows that:

(i)

v is incident to at most $k-2$ bounded 3-faces;

(ii)

v is incident to at most $k-3$ bounded 3-faces, if v has an incident 5-face.

Proof. 

If v is incident to $k-1$ bounded 3-faces, then there are four consecutive bounded faces forming a 4-cluster that is not a wheel, contrary to $G\in \mathcal{A}$. This proves (i). It follows from Lemmas 5 (iii) and (iv) that each 3-face in a $2^{+}$-cluster is not adjacent to a 5-face. Thus, each $5^{+}$-vertex incident to a 5-face must be incident to at least three $4^{+}$-faces. This proves (ii). □

Lemma 6

(Lemma 3.6 in [14]). $C(l_1,\dots ,l_k)$ is defined to be a cycle $C=x_1\dots x_m$ with k internal chords such that $x_1$ is their common endpoint and $V\left(C\right)\cap V\left(C_0\right)=\varnothing$. Suppose $x_2$ or $x_m$ is not the endpoint of any chords in $C.$ If $d\left(x_1\right)\le k+3,$ then some $i\in \{2,3,\dots ,m\}$ satisfies $d\left(x_i\right)\ge 5$ (See the proof in Lemma A4).

Lemma 7.

Let a 4-vertex v be incident to bounded faces $f_1,\dots ,f_4$ in cyclic order and let $F=B\left(f_1\right)\cup B\left(f_2\right)$, where $V\left(F\right)\cap V\left(C_0\right)=\varnothing .$ If $(d\left(f_1\right),d\left(f_2\right))=(3,3)$ or $(3,5)$, then there is a vertex $w\in V\left(F\right)-\left\{v\right\}$ such that $d\left(w\right)\ge 5.$

Proof. 

If $(d\left(f_1\right),d\left(f_2\right))=(3,3)$, it follows from Lemma 3 (ii) that $F=C(3,3)$. Moreover, F has exactly one chord, otherwise there is a separating 3-cycle, which contradicts Lemma 1.

If $(d\left(f_1\right),d\left(f_2\right))=(3,5)$, it follows from Lemma 3 (ii) that $F=C(3,5)$. Moreover, F has exactly one chord by Lemma 4.

The proof is complete by Lemma 6. □

Lemma 8.

Let v be a 5-vertex with incident bounded faces $f_1,\dots ,f_5$ in a cyclic order. Let $F=B_1\cup B_2\cup B_3$ where $B_i$ denote $B\left(f_i\right)$ and $V\left(F\right)\cap V\left(C_0\right)=\varnothing .$ If $(d\left(f_1\right),d\left(f_2\right),d\left(f_3\right))=(5,3,5),$ then there exists $w\in V\left(F\right)-\left\{v\right\}$ such that $d\left(w\right)\ge 5.$

Proof. 

Let $B_1=x_1{x}_2{x}_3{x}_4{x}_5,$$B_2=x_1{x}_5{x}_6,$ and $B_3=x_1{x}_6{x}_7{x}_8{x}_9$, where $x_1=v.$ It follows from Lemma 3 (ii) that $B_1\cup B_2$ is a $C(3,5)$ and $B_2\cup B_3$ is a $C(3,5)$. Suppose to the contrary that F is not a $C(5,3,5)$. Then, there is $i\in \{2,3,4\}$ and $j\in \{7,8,9\}$ such that $x_i=x_j$. If $i=2,$ then a 6-cylcle $x_1{x}_5{x}_6{x}_7{x}_8{x}_9$ has a triangular chord $x_1{x}_6$ and a chord $x_1{x}_j,$ contrary to Lemma 4. If $i=2,$ then a 6-cylcle $x_1{x}_5{x}_6{x}_7{x}_8{x}_9$, has a triangular chord $x_1{x}_6$ and a chord $x_5{x}_j,$ contrary to Lemma 4.

Suppose that $i=3.$ Note that a 6-cycle $C=x_1{x}_5{x}_6{x}_7{x}_8{x}_9$ is adjacent to a 3-cycle $x_1{x}_5{x}_6$ and a 5-cycle $x_1{x}_6{x}_7{x}_8{x}_9$. It suffices to show that C is adjacent to a 4-cycle to get a contradiction. If $x_3=x_7$, then C is adjacent to a 4-cycle $x_1{x}_2{x}_7{x}_6$. If $x_3=x_8$, then C is adjacent to a 4-cycle $x_1{x}_2{x}_8{x}_9.$ If $x_3=x_9$, then C is adjacent to a 4-cycle $x_1{x}_5{x}_4{x}_9.$

Thus, $F=C(5,3,5)$. By Lemma 6, it remains to show that $x_2$ or $x_m$ is not an endpoint to a chord in $C,$ say $x_1{x}_2\cdots x_9.$ Suppose C has a chord $e=x_2{x}_i,$ otherwise the desired condition is obtained. If $x_2{x}_9\in E\left(G\right)$, then we have separating 3-cycle $x_1{x}_2{x}_9,$ contrary to Lemma 1. By Lemma 4, we have $i\notin \{4,5,6\}.$ Then, $x_i=x_7$ or $x_8.$ By Lemma 4, $x_9$ is not adjacent to $x_6$ or $x_7.$ Thus, a chord of $C^{\prime }$ cannot have $x_9$ as its endpoint. □

Corollary 4.

Let v be a 4-vertex incident to bounded faces $f_1,\dots ,f_4$ in cyclic order, where $f_1$ is an inner 5-face, $f_2$ is an inner 3-face, $f_3$ is an inner 5-face, and $f_4$ is an arbitrary face. If $f_3$ is a poor 5-face, then $f_1$ is a rich 5-face or an improper semi-rich 5-face.

Proof. 

Let $B_1=x_1{x}_2{x}_3{x}_4{x}_5,$$B_2=x_1{x}_5{x}_6,$ and $B_3=x_1{x}_6{x}_7{x}_8{x}_9$, where $x_1=v.$ Let $f_3$ be a poor 5-face. Then, $x_1,x_6,x_7,x_8,$ and $x_9$ are 4-vertices. By Lemma 7, $x_5$ is a $5^{+}$-vertex. If $x_2,x_3,$ or $x_4$ is a $5^{+}$-vertex, then $f_1$ is a rich 5-face. Now suppose that $x_2,x_3,$ and $x_4$ are 4-vertices. If $f_4$ is a not a 3-face, then $f_1$ is an improper semi-rich 5-face. If $f_4$ is a 3-face, then $x_2$ is a $5^{+}$-vertex by considering $f_1$ and $f_4$ into Lemma 7, a contradiction. □

4. Discharging Process

In this section, we use the discharging procedure to get a contradiction and complete the proof of Theorem 2.

For each vertex and bounded face $x\in V\left(G\right)\cup F\left(G\right)$, let an initial charge of x be $\mu \left(x\right)=d\left(x\right)-4$ and let $\mu \left(D\right)=d\left(D\right)+4=7$ where D is the unbounded face. By Euler’s Formula, ${\sum }_{x\in V\cup F}\mu \left(x\right)=0$. Let ${\mu }^{*}\left(x\right)$ be the charge after the discharge procedure of $x\in V\cup F.$ To get a contradiction, we prove that ${\mu }^{*}\left(x\right)\ge 0$ for each $x\in V\left(G\right)\cup F\left(G\right)$ and ${\mu }^{*}\left(D\right)>0.$

Let $w(x\to f)$ be the transferred charge from x to a face f where x is a vertex or a face.

The discharging rules:

(R1) Let v be a 5-vertex where $v\notin V\left(C_0\right)$ and f be an incident 3-face of v.

$w(v\to f)=\left\{\begin{array}{cc}\frac{1}{2},\hfill & \mathrm{if}v\mathrm{is}\mathrm{incident}\mathrm{to}\mathrm{some}5-\mathrm{faces},\hfill \\ \frac{1}{7},\hfill & \mathrm{if}v\mathrm{is}\mathrm{not}\mathrm{incident}\mathrm{to}\mathrm{any}5-\mathrm{faces}\mathrm{and}\hfill \\ & f\mathrm{is}\mathrm{not}\mathrm{adjacent}\mathrm{to}\mathrm{any}\mathrm{incident}3-\mathrm{faces}\mathrm{of}v,\hfill \\ \frac{3}{7},\hfill & \mathrm{if}v\mathrm{is}\mathrm{not}\mathrm{incident}\mathrm{to}\mathrm{any}5-\mathrm{faces}\mathrm{and}\hfill \\ & f\mathrm{is}\mathrm{adjacent}\mathrm{to}\mathrm{exactly}\mathrm{one}\mathrm{incident}3-\mathrm{face}\mathrm{of}v.\hfill \end{array}\right.$

(R2) Let v be a $6^{+}$-vertex where $v\notin V\left(C_0\right)$ and f be an incident 3-face of v.

$w(v\to f)=\left\{\begin{array}{cc}\frac{2}{3},\hfill & \mathrm{if}v\mathrm{is}\mathrm{incident}\mathrm{to}\mathrm{some}5-\mathrm{faces},\hfill \\ \frac{1}{2},\hfill & \mathrm{if}v\mathrm{is}\mathrm{not}\mathrm{incident}\mathrm{to}\mathrm{any}5-\mathrm{faces}.\hfill \end{array}\right.$

Let g be a k-face with k incident vertices, say $v_1$, $v_2,\cdots ,v_k$ in cyclic order, and with k adjacent faces, say $f_1$, $f_2,\cdots ,f_k$ in cyclic order. Let $f_i$ be incident to $v_i$ and $v_{i+1}$ (i is taken modulo k).

(R3) Let g be a 4-face.

$w(g\to f_i)=\frac{1}{3}$ if $f_i$ is an inner 3-face.

(R4) Let g be a 5-face.

$w(g\to f_i)=\frac{1}{5}$ if $f_i$ is a 4-face.

• Let g be an inner poor 5-face.
  $w(g\to f_i)=\frac{1}{5}$ if $f_i$ is an inner 3-face.
• Let g be an inner proper semi-rich 5-face.
  $w(g\to f_i)=\frac{1}{3}$ if $f_i$ is an inner 3-face where both $v_i$ and $v_{i+1}$ are 4-vertices.
• Let g be an inner rich 5-face or an inner improper semi-rich 5-face.
  $w(g\to f_i)=\left\{\begin{array}{cc}\frac{1}{6},\hfill & \mathrm{if}{f}_i\mathrm{is}\mathrm{an}\mathrm{inner}3-\mathrm{face}\mathrm{where}\mathrm{exactly}\mathrm{one}\mathrm{of}{v}_i\mathrm{and}{v}_{i+1}\mathrm{is}\mathrm{a}4-\mathrm{vertex}.\hfill \\ \frac{1}{3},\hfill & \mathrm{if}{f}_i\mathrm{is}\mathrm{an}\mathrm{inner}3-\mathrm{face}\mathrm{where}\mathrm{both}{v}_i\mathrm{and}{v}_{i+1}\mathrm{are}4-\mathrm{vertices}.\hfill \end{array}\right.$
• Let g be an extreme 5-face.
  $w(g\to f_i)=\frac{2}{3}$ if $f_i$ is an inner 3-face.

(R5) Let g be a k-face where $k\ge 6$.

$w(g\to f_i)=\left\{\begin{array}{cc}\theta \left(f_i\right)+\chi \left(f_{i+1}\right)\theta \left(f_{i+1}\right)+\chi \left(f_{i-1}\right)\theta \left(f_{i-1}\right),\hfill & \mathrm{if}{f}_i\mathrm{is}\mathrm{a}{4}^{-}-\mathrm{face},\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$ where $\chi \left(f_i\right)=\left\{\begin{array}{cc}\frac{1}{2},\hfill & \mathrm{if}{f}_i\mathrm{is}\mathrm{not}\mathrm{a}{4}^{-}-\mathrm{face},\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$ and $\theta \left(f_i\right)=\frac{d\left(g\right)-4}{d\left(g\right)}$ for each $i\in \{1,2,\cdots ,n\}$.

(R6) The unbounded face D incident to a vertex v receives charge $\mu \left(v\right)$ from v but gives 1 to each of its intersecting 3-faces and 5-faces.

It follows from (R6) that ${\mu }^{*}\left(v\right)=0$ for every $v\in V\left(C_0\right).$ By this, we consider only a vertex v such that $v\notin V\left(C_0\right).$

CASE 1:v is a 5-vertex.

• v is incident to some 5-faces.
  Then, v has at most two incident 3-faces by Corollary 3. Thus, ${\mu }^{*}\left(v\right)\ge \mu \left(v\right)-2\times \frac{1}{2}=0$ by (R1).
• v is not incident to any 5-faces.
  It follows from Corollary 3 that v is incident to at most three 3-faces. Then, v has at most two incident 3-faces, which are adjacent to exactly one incident 3-face of v. Thus, ${\mu }^{*}\left(v\right)\ge \mu \left(v\right)-2\times \frac{3}{7}-\frac{1}{7}=0$ by (R1).

CASE 2:v is a $6^{+}$-vertex.

• v is incident to some 5-faces.
  It follows from Corollary 3 that v is incident to not more than $d\left(v\right)-3$ of 3-faces. Thus, ${\mu }^{*}\left(v\right)\ge \mu \left(v\right)-(d\left(v\right)-3)\times \frac{2}{3}=(d\left(v\right)-4)-(\frac{2d\left(v\right)}{3}-2)=\frac{d\left(v\right)}{3}-2\ge 0$ by (R2) and $d\left(v\right)\ge 6$.
• v is not incident to any 5-faces.
  It follows from Corollary 3 that v is incident to at most $d\left(v\right)-2$ 3-faces. Thus, ${\mu }^{*}\left(v\right)\ge \mu \left(v\right)-(d\left(v\right)-2)\times \frac{1}{2}=(d\left(v\right)-4)-(\frac{d\left(v\right)}{2}-1)=\frac{d\left(v\right)}{2}-3\ge 0$ by (R2) and $d\left(v\right)\ge 6$.

For a 3-face in an i-cluster $H_i$, we consider the total of charges in the same cluster. That is $\mu \left(H_i\right)=-i$ and we show that ${\mu }^{*}\left(H_i\right)\ge 0$ instead.

CASE 3:f is a 3-face in an i-cluster, say $H_i$ where $|V\left(H_i\right)\cap V\left(C_0\right)|\ge 1$.

• If $|V\left(H_1\right)\cap V\left(C_0\right)|\ge 1$, then ${\mu }^{*}\left(H_1\right)\ge \mu \left(H_1\right)+1=0$ by (R6).
• If $|V\left(H_2\right)\cap V\left(C_0\right)|=1$, then each adjacent face of $H_2$ is a $6^{+}$-face by Lemma 5 (iii). Thus, ${\mu }^{*}\left(H_2\right)\ge \mu \left(H_2\right)+1+4\times \frac{1}{3}>0$ by (R5) and (R6).
• If $|V\left(H_2\right)\cap V\left(C_0\right)|\ge 2$, then each 3-face in $H_2$ is an extreme 3-face. Thus, ${\mu }^{*}\left(H_2\right)\ge \mu \left(H_2\right)+2\times 1=0$ by (R6).
• If $|V\left(H_3\right)\cap V\left(C_0\right)|=1$, then each adjacent face of $H_3$ is a $7^{+}$-face by Lemma 5 (iv). Thus, ${\mu }^{*}\left(H_3\right)\ge \mu \left(H_3\right)+1+5\times \frac{3}{7}>0$ by (R5) and (R6).
• If $|V\left(H_3\right)\cap V\left(C_0\right)|=2$, then $H_3$ is adjacent to at least four $7^{+}$-faces by Lemma 5 (iv). Moreover, there are at least two extreme 3-faces in $H_3$. Thus, ${\mu }^{*}\left(H_3\right)\ge \mu \left(H_3\right)+2+4\times \frac{3}{7}>0$ by (R5) and (R6).
• If $|V\left(H_3\right)\cap V\left(C_0\right)|=3$, then each 3-face in $H_3$ is an extreme 3-face. Thus, ${\mu }^{*}\left(H_3\right)\ge \mu \left(H_3\right)+3\times 1=0$ by (R6).
• If $|V\left(H_4\right)\cap V\left(C_0\right)|=1$, then there are two extreme 3-faces in $H_4$ and each adjacent face of $H_4$ is a $7^{+}$-face by Lemma 5 (iv). If each vertex in $V\left(H_4\right)-V\left(C_0\right)$ is a 4-vertex, we have ${\mu }^{*}\left(H_4\right)\ge \mu \left(H_4\right)+2+2\times \frac{9}{14}+2\times \frac{6}{7}>0$ by (R5) and (R6). Otherwise, there is a vertex in $V\left(H_4\right)-V\left(C_0\right)$, which is not a 4-vertex, then we have ${\mu }^{*}\left(H_4\right)\ge \mu \left(H_4\right)+2+6\times \frac{3}{7}>0$ by (R1), (R2), (R5), and (R6).
• If $|V\left(H_4\right)\cap V\left(C_0\right)|=2$, then there are at least three extreme 3-faces in $H_4$. Moreover, $H_3$ is adjacent to at least three $7^{+}$-faces by Lemma 5 (iv). Thus, ${\mu }^{*}\left(H_4\right)\ge \mu \left(H_4\right)+3\times 1+3\times \frac{3}{7}>0$ by (R5) and (R6).
• If $|V\left(H_4\right)\cap V\left(C_0\right)|=3$, then each 3-face in $H_4$ is an extreme 3-face. Thus, ${\mu }^{*}\left(H_4\right)\ge \mu \left(H_4\right)+4\times 1=0$ by (R6).

CASE 4:f is an inner 3-face in a 1-cluster.

Let $v_1$, $v_2,v_3$ be three incident vertices in cyclic order and $f_1$, $f_2,f_3$ be three adjacent faces in cyclic order. Moreover, let $f_i$ be incident to $v_i$ and $v_{i+1}$ (i is taken modulo 3) (See Figure 1).

Subcase 4.1:f is not adjacent to any 5-faces.

Thus, ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)+3\times \frac{1}{3}=0$ by (R3) and (R5).

Next, we consider that f is adjacent to some 5-faces in Subcases 4.2 to 4.5. It follows from Lemma 5 (ii) and the assumption of Case 4 that f is not adjacent to a $4^{-}$-faces.

Subcase 4.2: An inner 3-face f is adjacent to some extreme 5-faces.

WLOG, let $f_1$ be an extreme 5-face. Then, $w(f_1\to f)=\frac{2}{3}$ by (R4).

• $f_i$ is not an inner 5-face where $i=2$ or 3.
  Then, $f_i$ is an extreme 5-face or a $6^{+}$-face. Thus, $w(f_i\to f)\ge \frac{1}{3}$ by (R4) and (R5). Therefore, ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)+\frac{2}{3}+\frac{1}{3}=0.$
• $f_2$ and $f_3$ are inner 5-faces.
  - If $v_i$ is a $5^{+}$-vertex for some $i\in \{1,2,3\}$, then $w(v_i\to f)=\frac{1}{2}$ by (R1) and (R2). Thus, ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)+\frac{2}{3}+\frac{1}{2}>0$.
  - If $v_i$ is a 4-vertex for each $i\in \{1,2,3\}$, then $w(f_2\to f)\ge \frac{1}{5}$ and $w(f_3\to f)\ge \frac{1}{5}$ by (R4). Thus, ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)+\frac{2}{3}+2\times \frac{1}{5}>0.$

We now consider the cases that each adjacent 5-face of f is not an extreme 5-face.

Subcase 4.3:f is a poor 3-face.

It follows from Lemma 7 that $f_i$ is not a poor 5-face for each $i\in \{1,2,3\}$. Thus, ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)+3\times \frac{1}{3}=0$ by (R4) and (R5).

Subcase 4.4:f is a semi-rich 3-face.

Let $v_1$ be a $5^{+}$-vertex. By symmetry, we only consider two following cases.

• $f_2$ is a poor 5-face.
  Then $w(f_2\to f)\ge \frac{1}{5}$ by (R4). Note that if $f_i$ is an improper semi-rich 5-face, a rich 5-face, or a $6^{+}$-face where $i\in \{1,3\}$, then $w(f_i\to f)\ge \frac{1}{6}$ by (R4) and (R5).
  - If $f_i$ is a 5-face for $i=1$ or 3, then $f_i$ is an improper semi-rich 5-face or a rich 5-face by Corollary 4. It follows that $w(v_1\to f)\ge \frac{1}{2}$ by (R1) and (R2). Thus, ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)+2\times \frac{1}{6}+\frac{1}{5}+\frac{1}{2}>0$.
  - If $f_1$ and $f_3$ are $6^{+}$-faces, then $w(v_1\to f)\ge \frac{1}{7}$ by (R1) and (R2). Thus, ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)+2\times \frac{1}{3}+\frac{1}{5}+\frac{1}{7}>0$.
• $f_2$ is a $5^{+}$-face but not a poor 5-face.
  Then $w(f_2\to f)\ge \frac{1}{3}$ by (R4) and (R5). If $f_1$ and $f_3$ are $6^{+}$-faces, then ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)+3\times \frac{1}{3}=0$ by (R5). If $v_1$ is a $6^{+}$-vertex and $f_1$ or $f_3$ is a 5-face, then ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)+\frac{1}{3}+\frac{2}{3}=0$ by (R2). Thus, it remains to check the case that $f_1$ or $f_3$ is a 5-face and $v_1$ is a 5-vertex. Note that $w(v_1\to f)\ge \frac{1}{2}$ by (R1).
  - If $f_1$ and $f_3$ are 5-faces, then $f_i$ is a rich 5-face for $i=1$ or 3 by Lemma 8. It follows that $w(f_i\to f)=\frac{1}{6}$ by (R4). Thus, ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)+\frac{1}{3}+\frac{1}{2}+\frac{1}{6}=0$.
  - If $f_1$ is a 5-face and $f_3$ is a $6^{+}$-face, then $w(f_3\to f)\ge \frac{1}{3}$ by (R5). Thus, ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)+2\times \frac{1}{3}+\frac{1}{2}=0$.

Subcase 4.5:f is an inner rich 3-face.

Let $v_1$ and $v_2$ be $5^{+}$-vertices. Recall that $f_1,f_2$, and $f_3$ are inner $5^{+}$-faces and at least one of them is a 5-face. By symmetry, we only consider two following cases.

• $f_1$ is a 5-face or $f_2$ and $f_3$ are 5-faces.
  That makes $v_1$ and $v_2$ incident to some 5-faces. Then, $w(v_1\to f)\ge \frac{1}{2}$ and $w(v_2\to f)\ge \frac{1}{2}$ by (R1) and (R2). Thus, ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)+2\times \frac{1}{2}=0$.
• $f_1$ is a $6^{+}$-face and either $f_2$ or $f_3$ is a $6^{+}$-face.
  WLOG, let $f_2$ be a 5-face. That makes $v_2$ incident to some 5-faces. Then $w(f_1\to f)\ge \frac{1}{3}$ and $w(f_3\to f)\ge \frac{1}{3}$ by (R5) and $w(v_2\to f)\ge \frac{1}{2}$ by (R1) and (R2). Thus, ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)+2\times \frac{1}{3}+\frac{1}{2}>0$.

CASE 5:f is a 3-face in a 2-cluster $H_2$ where $|V\left(H_2\right)\cap V\left(D\right)|=0$.

Let $H_2$ be a 4-cycle $v_1{v}_2{v}_3{v}_4$ with a chord $v_1{v}_3$. Let $f_1$, $f_2,f_3,f_4$ be four adjacent faces of $H_2$ in cyclic order. Moreover, let $f_i$ be incident to $v_i$ and $v_{i+1}$ (i is taken modulo 4) (See Figure 2). It follows from Lemma 5 (iii) that $f_1$, $f_2,f_3$, and $f_4$ are $6^{+}$-faces. By symmetry, we only consider two following cases.

• $v_1$ and $v_3$ are 4-vertices.
  Then $w(f_i\to H_2)\ge \frac{1}{2}$ for $i\in \{1,2,3,4\}$ by (R5). Thus, ${\mu }^{*}\left(H_2\right)\ge \mu \left(H_2\right)+4\times \frac{1}{2}=0$.
• $v_1$ is a $5^{+}$-vertex and $v_3$ is a $4^{+}$-vertex.
  Then $w(v_1\to H_2)\ge 2\times \frac{3}{7}$ by (R1) and (R2), and $w(f_i\to H_2)\ge \frac{1}{3}$ for $i\in \{1,2,3,4\}$ by (R5). Thus, ${\mu }^{*}\left(H_2\right)\ge \mu \left(H_2\right)+4\times \frac{1}{3}+2\times \frac{3}{7}>0$.

CASE 6:f is a 3-face in a 3-cluster $H_3$ where $|V\left(H_3\right)\cap V\left(D\right)|=0$.

Let $H_3$ be a 5-cycle $v_1{v}_2{v}_3{v}_4{v}_5$ with two chords $v_1{v}_3$ and $v_1{v}_4$. Let $f_1$, $f_2,f_3,f_4,f_5$ be five adjacent faces of $H_3$ in cyclic order. Moreover, let $f_i$ be incident to $v_i$ and $v_{i+1}$ (i is taken modulo 5). Note that $f_1$ and $f_5$ may be the same face (See Figure 3). It follows from Lemma 5 (iv) that $f_1$, $f_2,f_3,f_4$, and $f_5$ are $7^{+}$-faces. By symmetry, we only consider the two following cases.

• $v_3$ and $v_4$ are 4-vertices.
  Then $w(f_1\to H_3)\ge \frac{3}{7}$ and $w(f_5\to H_3)\ge \frac{3}{7}$ by (R5), $w(f_2\to H_3)\ge \frac{9}{14}$ and $w(f_4\to H_3)\ge \frac{9}{14}$ by (R5), and $w(f_3\to H_3)=\frac{6}{7}$ by (R5). Thus, ${\mu }^{*}\left(H_3\right)\ge \mu \left(H_3\right)+2\times \frac{3}{7}+2\times \frac{9}{14}+\frac{6}{7}=0$.
• $v_3$ is a $5^{+}$-vertex and $v_4$ is a $4^{+}$-vertex.
  Then $w(v_3\to H_3)\ge 2\times \frac{3}{7}$ by (R1) and (R2), and $w(f_i\to H_3)\ge \frac{3}{7}$ for $i\in \{1,2,3,4,5\}$ by (R5). Thus, ${\mu }^{*}\left(H_3\right)\ge \mu \left(H_3\right)+7\times \frac{3}{7}=0.$

CASE 7:f is a 3-face in a 4-cluster $H_4$ where $|V\left(H_4\right)\cap V\left(D\right)|=0$.

Let $H_4$ be the wheel $W_5$ where $v_5$ is a hub and $v_1$,$v_2,v_3,$ and $v_4$ are external vertices in cyclic order. Let $f_1$, $f_2,f_3,f_4$ be four adjacent faces of $H_4$ in cyclic order. Moreover, let $f_i$ be incident to $v_i$ and $v_{i+1}$ (i is taken modulo 4) (See Figure 4). By Lemma 5 (iv), $f_1$, $f_2,f_3$, and $f_4$ are $7^{+}$-faces. Moreover, at least two vertices in $\{v_1,v_2,v_3,v_4\}$ are $5^{+}$-vertices by Lemma 7. By symmetry, we only consider the three following cases.

• $v_1$ and $v_2$ are $5^{+}$-vertices and $v_3$ and $v_4$ are 4-vertices.
  Then $w(f_1\to H_4)\ge \frac{3}{7}$, $w(f_2\to H_4)\ge \frac{9}{14}$, $w(f_4\to H_4)\ge \frac{9}{14}$, $w(f_3\to H_4)=\frac{6}{7}$ by (R5), $w(v_1\to H_4)\ge 2\times \frac{3}{7}$ and $w(v_2\to H_4)\ge 2\times \frac{3}{7}$ by (R1) and (R2). Thus, ${\mu }^{*}\left(H_4\right)\ge \mu \left(H_4\right)+2\times \frac{9}{14}+\frac{6}{7}+5\times \frac{3}{7}>0$.
• $v_1$ and $v_3$ are $5^{+}$-vertices and $v_2$ and $v_4$ are 4-vertices.
  Then $w(f_i\to H_4)\ge \frac{9}{14}$ for $i\in \{1,2,3,4\}$ by (R5), and $w(v_1\to H_4)\ge 2\times \frac{3}{7}$ and $w(v_3\to H_4)\ge 2\times \frac{3}{7}$ by (R1) and (R2). Thus, ${\mu }^{*}\left(H_4\right)\ge \mu \left(H_4\right)+4\times \frac{9}{14}+4\times \frac{3}{7}>0$.
• $v_1,v_2$, and $v_3$ are $5^{+}$-vertices and $v_4$ is a $4^{+}$-vertex.
  Then $w(f_i\to H_4)\ge \frac{3}{7}$ for $i\in \{1,2,3,4\}$ by (R5) and $w(v_i\to H_4)\ge 2\times \frac{3}{7}$ for $i\in \{1,2,3\}$ by (R1) and (R2). Thus, ${\mu }^{*}\left(H_4\right)\ge \mu \left(H_4\right)+10\times \frac{3}{7}>0$.

CASE 8:f is a 4-face adjacent to an inner 3-face, say h.

Since h is an inner 3-face, we have $\left|B\right(f)\cap B(D\left)\right|\le 2$ where D is the unbounded 3-face. Consequently, there are at least two adjacent faces of f, which are not h and D. Moreover, they are $5^{+}$-faces by Lemma 5 (i). Thus ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)-\frac{1}{3}+2\times \frac{1}{5}>0$ by (R3), (R4), and (R5).

CASE 9:f is a 5-face.

• Let f be adjacent to some 4-faces.
  Then f is not adjacent to any 3-faces by Lemma 5 (ii). Thus, ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)-5\times \frac{1}{5}=0$ by (R4).
• Let f be an inner poor 5-face.
  Then ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)-5\times \frac{1}{5}=0$ by (R4).
• Let f be an inner semi-rich 5-face.
  - If f is a proper semi-rich 5-face, then $B\left(f\right)$ has three edges with two 4-endpoints. Thus, ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)-3\times \frac{1}{3}=0$ by (R4).
  - If f an improper semi-rich 5-face, then $B\left(f\right)$ has at most two edges with two 4-endpoints and at most two edges with exactly one $5^{+}$-endpoint. Thus, ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)-2\times \frac{1}{3}-2\times \frac{1}{6}=0$ by (R4).
• Let f be an inner rich 5-face.
  Then f has at least two incident $5^{+}$-vertices.
  If two incident $5^{+}$-vertices are not adjacent in $B\left(f\right)$, then $B\left(f\right)$ has at most one edge with two 4-endpoints. Thus, ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)-\frac{1}{3}-4\times \frac{1}{6}=0$ by (R4). It remains to consider the case that f has exactly two incident $5^{+}$-vertices and they are adjacent in $B\left(f\right).$ Then $B\left(f\right)$ has two edges with two 4-endpoints and two edges with exactly one $5^{+}$-endpoint. Thus, ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)-2\times \frac{1}{3}-2\times \frac{1}{6}=0$ by (R4).
• Let f be an extreme 5-face.
  Then f has at most an adjacent inner 3-face. Thus, ${\mu }^{*}\left(f\right)\ge \mu \left(f\right)+1-3\times \frac{2}{3}=0$ by (R4) and (R6).

CASE 10:f is an m-face where $m\ge 6$.

Then, by (R5) we have $w(f\to f_i)\le (1-2\chi \left(f_i\right))\theta \left(f_i\right)+\chi \left(f_{i+1}\right)\theta \left(f_{i+1}\right)+\chi \left(f_{i-1}\right)\theta \left(f_{i-1}\right)$.

$$\begin{array}{cc}\hfill {\mu }^{*}\left(f\right)& =\mu \left(f\right)-\sum _{i=1}^{m}w(f\to f_i)\hfill \\ \hfill & \ge \mu \left(f\right)-\sum _{i=1}^m((1-2\chi \left(f_i\right))\theta \left(f_i\right)+\chi \left(f_{i+1}\right)\theta \left(f_{i+1}\right)+\chi \left(f_{i-1}\right)\theta \left(f_{i-1}\right))\hfill \\ \hfill & =\mu \left(f\right)-\sum _{i=1}^m(\theta \left(f_i\right)-2\chi \left(f_i\right)\theta \left(f_i\right)+2\chi \left(f_i\right)\theta \left(f_i\right))\hfill \\ \hfill & =m-4-m\left(\frac{m-4}{m}\right)\hfill \\ \hfill & =0.\hfill \end{array}$$

CASE 11: The unbounded face D.

Let the number of intersecting 3-faces and 5-faces of D be denoted by $f^{\prime }.$ Let $E(C_0,V\left(G\right)-C_0)$ denote the set of edges between $V\left(G\right)-C_0$ and $C_0$ where this set has size $e(C_0,V\left(G\right)-C_0).$ Then by (R6),

$$\begin{array}{cc}\hfill {\mu }^{*}\left(D\right)& =3+4+\sum _{v\in C_0}(d\left(v\right)-4)-f^{\prime }\hfill \\ \hfill & =1+\sum _{v\in C_0}(d\left(v\right)-2)-f^{\prime }\hfill \\ \hfill & =1+e(C_0,V\left(G\right)-C_0)-f^{\prime }.\hfill \end{array}$$

So we may consider that D sends charge 1 to each edge $e\in E(C_0,V\left(G\right)-C_0)$. So each intersecting 3-face and 5-face contains at least two edges in $E(C_0,V\left(G\right)-C_0)$. It follows that $e(C_0,V\left(G\right)-C_0)-f^{\prime }\ge 0$. Thus, ${\mu }^{*}\left(D\right)>0.$

This completes the proof.

5. Conclusions

We prove that every planar graph without 6-cycles simultaneously adjacent to 3-cycles, 4-cycles, and 5-cycles is DP-4-colorable. This result is a special case of two following open problems.

1. Every planar graph without i-cycles simultaneously adjacent to j-cycles, k-cycles, and l-cycles is DP-4-colorable for $\{i,j,k,l\}=\{3,4,5,6\}$.

2. Every planar graph without 3-, 4-, 5-, and 6-cycles that are pairwise adjacent is DP-4-colorable.