3-choosable planar graphs with some precolored vertices and no 5 − -cycles normally adjacent to 8 − -cycles
33-choosable planar graphs with some precolored vertices and no − -cycles normally adjacent to − -cycles Fangyao Lu Qianqian Wang Tao Wang ∗ Institute of Applied MathematicsHenan University, Kaifeng, 475004, P. R. China
Abstract
DP-coloring was introduced by Dvořák and Postle [J. Combin. Theory Ser. B 129 (2018) 38–54]as a generalization of list coloring. They used a "weak" version of DP-coloring to solve a longstandingconjecture by Borodin, stating that every planar graph without cycles of length to is -choosable. Liuand Li improved the result by showing that every planar graph without adjacent cycles of length at most is -choosable. In this paper, it is showed that every planar graph without − -cycles normally adjacentto − -cycles is -choosable. Actually, all these three papers give more stronger results by stating them inthe form of "weakly" DP- -coloring and color extension. All graphs in this paper are finite, undirected and simple. For a graph G , a list-assignment L assigns toeach vertex v a set L ( v ) of colors available at v . An L -coloring of G is a proper coloring φ of G such that φ ( v ) ∈ L ( v ) for all v ∈ V ( G ) . A list k -assignment L is a list-assignment such that | L ( v ) | ≥ k for all v ∈ V ( G ) . A graph G is k -choosable or list k -colorable if it has an L -coloring for any list k -assignment L . The list chromatic number or choice number χ (cid:96) ( G ) is the least integer k such that G is k -choosable.The Four Color Theorem states that every planar graph is -colorable. Grötzsch [4] showed that everyplanar graph without triangles is -colorable. Much more sufficient conditions for -colorability and -choosability are extensively studied. Thomassen [13] showed that every planar graph with girth at least fiveis -choosable. Borodin [1] conjectured that every planar graph without cycles of length to is -choosable.A widely used technique in ordinary vertex coloring is the identification of vertices, but this is not feasiblein general for list coloring because different vertices may have different lists. To overcome this difficulty,Dvořák and Postle [3] introduced DP-coloring, also called correspondence coloring, as a generalization of listcoloring. Definition 1.
Let G be a simple graph and L be a list-assignment for G . For each vertex v ∈ V ( G ) ,let L v = { v } × L ( v ) ; for each edge uv ∈ E ( G ) , let M uv be a matching between the sets L u and L v , andlet M := (cid:83) uv ∈ E ( G ) M uv . We call M a matching assignment . The matching assignment is called a k -matching assignment if L ( v ) = [ k ] for each v ∈ V ( G ) . A cover of G is a graph H L, M (simply write H )satisfying the following two conditions:(C1) the vertex set of H is the disjoint union of L v for all v ∈ V ( G ) ; ∗ Corresponding author: [email protected]; [email protected] a r X i v : . [ m a t h . C O ] D ec C2) the edge set of H is the matching assignment M .Note that the matching M uv is not required to be a perfect matching between the sets L u and L v , andpossibly it is empty. The induced subgraph H [ L v ] is an independent set for each vertex v ∈ V ( G ) . Definition 2.
Let G be a simple graph and H be a cover of G . An M -coloring of G is an independentset I in H such that |I ∩ L v | = 1 for each vertex v ∈ V ( G ) . The graph G is DP- k -colorable if for anylist assignment L ( v ) ⊇ [ k ] and any matching assignment M , it has an M -coloring. The DP-chromaticnumber χ DP ( G ) of G is the least integer k such that G is DP- k -colorable.DP-coloring is quite different from list coloring, for example each even cycle is -choosable but it is notDP- -colorable. Dvořák and Postle gave a relation between DP-coloring and list coloring.Let W = w w . . . w m with w m = w be a closed walk of length m in G , a matching assignment is inconsistent on W , if there exists c i ∈ L ( w i ) for i ∈ [ m ] such that ( w i , c i )( w i +1 , c i +1 ) is an edge in M w i w i +1 for i ∈ [ m − and c (cid:54) = c m . Otherwise, the matching assignment is consistent on W . Theorem 1.1 (Dvořák and Postle [3]) . A graph is k -choosable if and only if G is M -colorable for everyconsistent k -matching assignment M .With the aid of DP-coloring (actually "weakly" DP-coloring, see Theorem 1.3), Dvořák and Postle [3]solved the longstanding conjecture by Borodin. Theorem 1.2 (Dvořák and Postle [3]) . Every planar graph without cycles of length to is -choosable.An edge uv in G is straight in a k -matching assignment M if ( u, c )( v, c ) ∈ M uv satisfies c = c . Anedge uv in G is full in a k -matching assignment M if M uv is a perfect matching. Lemma 1.1 (Dvořák and Postle [3]) . Let G be a graph with a k -matching assignment M , and let K be asubgraph of G . If for every cycle Q in K , the assignment M is consistent on Q and all edges of Q are full,then we may rename L ( u ) for u ∈ V ( K ) to obtain a k -matching assignment M (cid:48) for G such that all edges of K are straight in M (cid:48) .In order to prove Theorem 1.2, they showed a stronger result as the following. Theorem 1.3 (Dvořák and Postle [3]) . Let G be a plane graph without cycles of length to . Let S bea set of vertices of G such that | S | ≤ or S consists of all vertices on a face of G . Let M be a -matchingassignment for G such that M is consistent on every closed walk of length three in G . If | S | ≤ , then every M -coloring φ of G [ S ] can be extended to an M -coloring ϕ of G .Two cycles are adjacent if they have at least one common edge. An (cid:96) -cycle and an (cid:96) -cycle are normallyadjacent if they form an ( (cid:96) + (cid:96) − -cycle with exactly one chord. In other words, two cycles are normallyadjacent if their intersection is K . Recently, Liu and Li [7] improved Theorem 1.3 to the following result byallowing cycles of length to but forbidding adjacent cycles of length at most . Theorem 1.4 (Liu and Li [7]) . Let G be a plane graph without adjacent cycles of length at most . Let S be a set of vertices of G such that | S | ≤ or S consists of all vertices on a face of G . Let M be a -matchingassignment for G such that M is consistent on every closed walk of length three in G . If | S | ≤ , then every M -coloring φ of G [ S ] can be extended to an M -coloring ϕ of G .This implies the -choosability of planar graphs without adjacent cycles of length at most . Theorem 1.5 (Liu and Li [7]) . Every planar graph without adjacent cycles of length at most is -choosable.2ig. 1: The abnormal cycleThe aim of this paper is to further improve Theorem 1.4 to the following result by allowing adjacent cyclesof length to and changing the condition on precolored vertices from faces to cycles. But before we statethe main theorem, it’s necessary to give a new concept. A cycle is normal if each vertex not on it has atmost two neighbors on the cycle. The abnormal − -cycles can only be the 12-cycle as in Fig. 1. A d -vertex, d + -vertex or d − -vertex is a vertex of degree d , at least d , or at most d respectively. Similar definitions canbe applied to faces and cycles. Theorem 1.6.
Let G be a plane graph without − -cycles normally adjacent to − -cycles. Let S be a set ofvertices of G such that | S | ≤ or S consists of all vertices on a normal cycle of G . Let M be a -matchingassignment for G such that M is consistent on every closed walk of length three in G . If | S | ≤ , then every M -coloring φ of G [ S ] can be extended to an M -coloring ϕ of G . Remark 1.
The graph in Fig. 1 is a plane graph without − -cycles normally adjacent to − -cycles. It isobserved that not every M -coloring of the -cycle can be extended to the whole graph. Thus, we requirethe condition that S consists of all vertices on a "normal" cycle.The following result is a direct consequence of Theorem 1.6, and it improves Theorem 1.5. Theorem 1.7.
Every planar graph without − -cycles normally adjacent to − -cycles is -choosable.The following three results are immediate consequences of Theorem 1.7. The first one generalizes the -colorability of such graphs by Luo, Chen and Wang [11], and the second one generalizes the -colorabilityof such graphs by Wang and Chen [14]. Corollary 1.8.
Every planar graph without , , , -cycles is -choosable. Corollary 1.9.
Every planar graph without , , , -cycles is -choosable. Corollary 1.10.
Every planar graph without , , , -cycles is -choosable. Remark 2.
Theorem 1.2, Theorem 1.5 and Theorem 1.7 are only for -choosable, but not for DP- -colorable.Since we require the "consistency" on every closed walk of length three, the graphs in Theorem 1.3, Theo-rem 1.4 and Theorem 1.6 are "weakly" DP- -colorable. It is interesting to know whether such graphs areDP- -colorable.Liu et al. gave some sufficient conditions for a planar graph to be DP- -colorable which extends the -choosability of such graphs. Theorem 1.11 (Liu et al. [10]) . A planar graph is DP- -colorable if it satisfies one of the following conditions:(1) G contains no { , , , } -cycles.(2) G contains no { , , } -cycles.(3) G contains no { , , , } -cycles. 34) G contains no { , , , } -cycles.(5) G contains no { , , } -cycles and no triangles at distance less than two. Theorem 1.12 (Liu et al. [9]) . If a and b are distinct values from { , , } , then every planar graph withoutcycles of lengths { , a, b, } is DP- -colorable.In the spirit of Bordeaux conditions, Yin and Yu [15] gave the following condition for planar graphs tobe DP-3-colorable. Theorem 1.13 (Yin and Yu [15]) . A planar graph is DP- -colorable if it satisfies one of the followingconditions:(1) G contains no { , } -cycles and no triangles at distance less than three.(2) G contains no { , , } -cycles and no triangles at distance less than two.DP- -colorable planar or toroidal graphs can be found in [2, 5, 6, 8]. Thomassen [12] showed that everyplanar graph is -choosable. Dvořák and Postle [3] observed that every planar graph is DP- -colorable.We need more notations in the next sections. Let G be a plane graph. The edges and vertices divide theplane into a number of faces . The unbounded face is called the outer face, and the other faces are called inner faces . An internal vertex is a vertex that is not incident with the outer face. An internal face is a face having no common vertices with the outer cycle. Let O be a cycle of a plane graph G , the cycle O divides the plane into two regions, the subgraph induced by all the vertices in the unbounded region isdenoted by ext( O ) , and the subgraph induced by all the vertices in the other region is denoted by int( O ) .If both int( O ) and ext( O ) contain at least one vertex, then we call the cycle O a separating cycle of G .The subgraph obtained from G by deleting all the vertices in ext( O ) is denoted by Int( O ) , and the subgraphobtained from G by deleting all the vertices in int( O ) is denoted by Ext( O ) . Let N be the set of inner faceshaving at least one common vertex with the outer face. In this section, we give a proof of the following main result.
Theorem 1.6.
Let G be a plane graph without − -cycles normally adjacent to − -cycles. Let S be a set ofvertices of G such that | S | ≤ or S consists of all vertices on a normal cycle of G . Let M be a -matchingassignment for G such that M is consistent on every closed walk of length three in G . If | S | ≤ , then every M -coloring φ of G [ S ] can be extended to an M -coloring ϕ of G . Proof.
Suppose that G is a minimal counterexample to Theorem 1.6. That is, there exists an M -coloringof G [ S ] that cannot be extended to an M -coloring of G such that | V ( G ) | is minimized. (1)Subject to (1), | E ( G ) | − | S | is minimized. (2)Subject to (1) and (2),the number of edges in the -matching assignment M is maximized. (3)By the structure of G , we immediately have the following result on − -cycles.4 emma 2.1. Every − -cycle has no chord. Moreover, every − -cycle is not adjacent to any -cycle.Next, we give some structural results on G . Some of the lemmas are almost the same with that in [3] and[7], but for completeness we give detailed proofs here. Lemma 2.2. (a) S (cid:54) = V ( G ) ;(b) G is -connected, and thus the boundary of every face is a cycle;(c) each vertex not in S has degree at least three;(d) either | S | = 1 or G [ S ] is an induced cycle of G ;(e) there is no separating normal k -cycle for ≤ k ≤ ;(f) G [ S ] is an induced cycle of G . Proof. (a) Suppose to the contrary that S = V ( G ) . Every M -coloring of G [ S ] is an M -coloring of G , acontradiction.(b) By the condition (1), G is connected. Suppose to the contrary that G has a cut-vertex w . We mayassume that G = G ∪ G and G ∩ G = { w } . By the hypothesis of the set S , either S ⊆ V ( G ) or S ⊆ V ( G ) . We may assume that S ⊆ V ( G ) . By the condition (1), the M -coloring φ of G [ S ] can beextended to an M -coloring φ of G , and φ ( w ) can be extended to an M -coloring φ of G . These twocolorings φ and φ together give an M -coloring of G whose restriction on G [ S ] is φ , a contradiction.(c) Suppose that there exists a vertex w not in S having degree at most two. By the condition (1), the M -coloring of G [ S ] can be extended to an M -coloring of G − w . Since w has degree at most two, there areat most two forbidden colors for w , thus we can extend the M -coloring of G − w to an M -coloring of G , acontradiction.(d) Suppose to the contrary that S = V ( Q ) and Q is a cycle with a chord uv . It is observed that the M -coloring of G [ S ] is also an M -coloring of the induced subgraph in G − uv . By the condition (2), the M -coloring φ of G [ S ] can be extended to an M -coloring of G − uv , and hence it is also an M -coloring of G ,a contradiction.(e) We first show that G [ S ] cannot be a separating cycle. Suppose to the contrary that G [ S ] is a separating(normal) cycle O . By the condition (1), the M -coloring φ of O can be extended to an M -coloring φ of Int( O ) , and another M -coloring φ of Ext( O ) . These two colorings φ and φ together give an M -coloringof G whose restriction on G [ S ] is φ , a contradiction.Thus, either | S | = 1 or S consists of all vertices on a face of G . Let Q be a separating normal k -cycle with ≤ k ≤ . Thus, we may assume that S ⊆ Ext( Q ) . By the condition (1), the M -coloring φ of G [ S ] can beextended to an M -coloring ϕ of Ext( Q ) . Similarly, the restriction of ϕ on G [ V ( Q )] can be extended to an M -coloring ϕ of Int( Q ) . These two colorings ϕ and ϕ together give an M -coloring of G whose restrictionon G [ S ] is φ , a contradiction.(f) According to (d), suppose to the contrary that S = { w } . We first assume that w is on an − -cycle Q .Without loss of generality, we may assume that Q is a shortest cycle containing w . Since Q is a shortest cyclecontaining w , the cycle Q is an induced cycle. By (e), we may assume that ext( Q ) = ∅ and Q is the outercycle. By (c) and Q is an induced cycle, thus every vertex on Q other than w has a neighbor in int( Q ) , whichimplies that int( Q ) (cid:54) = ∅ . By the condition (1), the M -coloring φ of { w } can be extended to an M -coloring φ of Q . By the condition (2), the M -coloring φ of Q can be further extended to an M -coloring of G , acontradiction. 5 a) w x yzw (b) xw yw z (c) w x y zw (d) Fig. 2: A -face is adjacent to an − -face, where the blue cycle bounds a -face and the red cycle bounds an − -faceSo we may assume that every cycle containing w has length at least . Let w be incident with a face w ww . . . w . Let G (cid:48) be obtained from G by adding a chord w w in the face, let S (cid:48) = { w, w , w } and letthe -matching assignment M (cid:48) for G (cid:48) be obtained from M by setting the matching corresponding to w w is edgeless. We can easily check that G (cid:48) is a planar graph without − -cycles normally adjacent to − -cycles.By the condition (1), the M -coloring φ of { w } can be extended to an M (cid:48) -coloring φ of G (cid:48) [ S (cid:48) ] . By thecondition (2), the M (cid:48) -coloring φ of G (cid:48) [ S (cid:48) ] can be further extended to an M (cid:48) -coloring ϕ of G (cid:48) . It is observedthat ϕ is an M -coloring of G , a contradiction.For convenience, we can redraw the graph G such that G [ S ] is the outer cycle C of G . Lemma 2.3.
There is no − -face adjacent to − -face. Proof.
Recall that every face is bounded by a cycle. Assume that f is an − -face w w w . . . w and it isadjacent to a − -face g . By Lemma 2.1, it suffices to consider that g is a - or -face.Suppose that g = uvw w is a -face. Since there is no − -cycle normally adjacent to − -cycle, we havethat either u or v is on f . By symmetry, we may assume that u is on f . Recall that every − -face is boundedby a cycle and this cycle has no chord, so u = w and v is not on f . It is observed that w is a -vertex andit must be on the outer cycle C . It follows that either f or g is the outer face. If f is the outer face, then v is an internal vertex and it has a neighbor not on C (since w vw . . . w has no chord by Lemma 2.1), thusthere is a separating − -cycle w vw . . . w , a contradiction. Similarly, if g is the outer face, then there isan internal vertex on f having a neighbor not on C , thus there is a separating − -cycle containing w vw , acontradiction.Suppose that g is a -face. Since every − -cycle has no chord, there are only four cases (up to symmetry)for the local structures, see Fig. 2. Since there is no − -cycle normally adjacent to an − -cycle, the first casewill not occur. For the other three cases, we first assume that x is an internal vertex. Since every internalvertex has degree at least three, x has a neighbor x (cid:48) other than w and y . It is observed that x is on a − -cycle O x not containing w . If x (cid:48) is on O x , then xx (cid:48) is a chord of O x , but this contradicts Lemma 2.1; if x (cid:48) is not on O x , then O x is a separating − -cycle, this contradicts Lemma 2.2(e). So we may assume that x is on the outer cycle. In the second and third cases, w is a -vertex, so it is on the outer cycle, and g must be the outer face. In the fourth case, by the symmetry of x and z , z is on the outer cycle, and g is theouter face. Therefore, g is the outer face in the last three cases, and there is an internal vertex on f havinga neighbor not on g , and then there is a separating − -cycle containing w xy , a contradiction. Lemma 2.4. If ww w is a triangle and w, w are internal -vertices, then all the edges in { ww , ww , w w } are full. 6 roof. Suppose to the contrary that at least one of ww , ww and w w is not full. By applying Lemma 1.1 to { ww , ww } , we may assume that ww and ww are straight in M . Let M (cid:48) be a new -matching assignmentfor G by setting M (cid:48) e = M e for each e / ∈ { ww , ww , w w } and all edges in { ww , ww , w w } are straightand full. Note that ww and ww are straight in M , thus M ww ⊆ M (cid:48) ww and M ww ⊆ M (cid:48) ww . Since allthe edges in { ww , ww , w w } are full in M (cid:48) but not in M , the number of edges in M (cid:48) is greater than thatin M . Since there is no adjacent triangles, every closed walk of length three is consistent in M (cid:48) . By thecondition (3), the M -coloring φ (also M (cid:48) -coloring) of the outer cycle C can be extended to an M (cid:48) -coloring φ (cid:48) of G , but φ (cid:48) is not an M -coloring of G by our assumption. Note that M e ⊆ M (cid:48) e for any e (cid:54) = w w , so we mayassume that φ (cid:48) ( w ) = 1 , φ (cid:48) ( w ) = 2 and ( w , w , ∈ M w w . If ( w , has an incident edge in M ww and ( w , has an incident edge in M ww , then the closed walk ww w w is not consistent in M , a contradiction.If ( w , has no incident edge in M ww , then we can modify φ (cid:48) to obtain an M -coloring of G by recoloring w and w in order, a contradiction. So we may assume that ( w , has an incident edge in M ww and ( w , hasno incident edge in M ww . Since ww is straight in M , we have that ( w , w, ∈ M ww . Furthermore, wemay assume that ( w, has no incident edge in M ww , otherwise the closed walk w ww w is not consistentin M . Now, we can obtain a new -matching assignment M ∗ for G by adding an edge ( w, w , to M .By the hypothesis, ww is only contained in the triangle ww w , so the addition of ( w, w , does notmake M ∗ inconsistent on closed -walk, but this contradicts the condition (3). Lemma 2.5.
Let w , w , w , w , w be five consecutive vertices on a + -face. If w , w , w , w are all -vertices and w w is on -face ww w , then at least one vertex in { w , w , w , w } is on the outer cycle C . Proof.
Suppose to the contrary that none of { w , w , w , w } is on the outer cycle C . Let w (cid:48) be the neighborof w other than w , w , and let H = G − { w , w , w , w } . It is observed that w , w , w , w , w , w and w (cid:48) are seven distinct vertices. We claim that the distance between w and w (cid:48) is at least nine in H . Let P be ashortest path between w and w (cid:48) in H . It is observed that Q = P ∪ w w w w w (cid:48) is a cycle. If w is on thepath P , then P [ w , w ] ∪ w w w and P [ w, w (cid:48) ] ∪ ww w w (cid:48) are all cycles, which implies that these two cycleshave length at least nine and | P | ≥ (9 −
2) + (9 −
3) = 13 . If w is not on the path P , then Q is a separatingnormal cycle (note that w and w are in different sides of the cycle Q ) and it has length at least , whichimplies that | P | = |Q| − ≥ − .By Lemma 2.4 and Lemma 1.1, we may assume that all the edges incident with the vertices in { w , w , w } are straight. Let G (cid:48) be the graph obtained from H by identifying w and w (cid:48) , and let M (cid:48) be the restrictionof M on E ( G (cid:48) ) . Since the distance between w and w (cid:48) is at least nine in H , the graph G (cid:48) has no loop, nomultiple edge and no new − -cycle, thus G (cid:48) is a simple planar graph without − -cycles normally adjacentto − -cycles. Moreover, C is also a normal cycle of G (cid:48) and it has no chord in G (cid:48) . This implies that φ is an M (cid:48) -coloring of G (cid:48) [ S ] . Since | V ( G (cid:48) ) | < | V ( G ) | , the M (cid:48) -coloring φ of G (cid:48) [ S ] can be extended to an M (cid:48) -coloring ϕ of G (cid:48) . Since w and w are all -vertices, we can extend ϕ to w and w in order. Recall that all theedges incident with vertices in { w , w , w } are straight, thus w and w have distinct colors, and then wecan further extend the coloring to w and w , a contradiction.Let w be a vertex on the outer cycle C , and let w , w , . . . , w k be consecutive neighbors in a cyclic order.If f is a face in N incident with ww i and ww i +1 , but neither ww i nor ww i +1 is an edge of C , then we call f a special face at w . An internal -vertex is bad if it is incident with a non-special -face, light if it isincident with a -face or a -face or a special -face, good if it is neither bad nor light (i.e., it is incidentwith three + -faces). According to Lemma 2.5, we have the following result on bad vertices. Lemma 2.6.
There is no five consecutive bad vertices on a + -face.7 emma 2.7. If a -face in N has at least two common vertices with C , then it has exactly two commonvertices with C , and these two vertices are consecutive on the -face. Proof.
Suppose that f = w w w w is a -face in N that has at least three common vertices with C . ByLemma 2.2(f), C is an induced cycle. So we may assume that w , w , w are three consecutive vertices on C and w is an internal vertex. By Lemma 2.2(c), w has another neighbor w (cid:48) other than w and w . If w (cid:48) is an internal vertex, then there exists a separating normal − -cycle, a contradiction. If w (cid:48) is on the outercycle C , then the -cycle w w w w is normally adjacent to an − -cycle, a contradiction.Suppose that f is a -face in N that has exactly two common vertices with C . If these two vertices arenot consecutive on the -face, then there exists a separating normal − -cycle, a contradiction.We give the initial charge µ ( v ) = deg( v ) − for any v ∈ V ( G ) , µ ( f ) = deg( f ) − for any face f ∈ F ( G ) other than outer face D , and µ ( D ) = deg( D ) + 4 for the outer face D . By the Euler formula, the sum of theinitial charges is zero. That is, (cid:88) v ∈ V ( G ) (cid:0) deg( v ) − (cid:1) + (cid:88) f ∈ F ( G ) \ D (cid:0) deg( f ) − (cid:1) + (deg( D ) + 4) = 0 . (4)Next, we give the discharging rules to redistribute the charges, preserving the sum, such that the final chargeof every element in V ( G ) ∪ F ( G ) is nonnegative, and at least one element in V ( G ) ∪ F ( G ) has positive finalcharge. This leads a contradiction to complete the proof. R1 Each non-special -face receives from each incident internal vertex. R2 Each bad vertex receives from each incident + -face; each light vertex receives from each incident + -face; each good vertex receives from each incident face. R3 Let w be an internal -vertex. If it is incident with exactly two non-special -faces, then it receives from each incident + -face. If it is incident with exactly one non-special -face f and three + -faces, then it receives from the + -face which is not adjacent to f . If it is incident with exactly onenon-special -face f and exactly two + -faces, then it receives from each incident + -face. R4 Each -vertex on the outer cycle C receives from the incident face in N and from the outer face. R5 Each -vertex on the outer cycle C receives from the outer face and sends to incident − -face in N and to each incident k -face in N , where ≤ k ≤ . R6 Each -vertex on the outer cycle C receives from the outer face, and sends to each incident special − -face, to each of the other incident − -face in N . R7 Each + -vertex on the outer cycle C receives from the outer face, and sends to each incident special − -face and to each of the other incident face in N . Lemma 2.8.
Every face other than D has nonnegative final charge. Proof.
According to the discharging rules, inner -faces never give charges. If f is a special -face, then µ (cid:48) ( f ) = 3 − by R6 and R7 . If f is a non-special -face having no vertex on the outer cycle C , then µ (cid:48) ( f ) = 3 − × = 0 by R1 . If f is a non-special -face having two vertices on the outer cycle C , then f has a common edge with the outer face by Lemma 2.2(f), and then µ (cid:48) ( f ) ≥ − × = 0 by R1 , R5 , R6 and R7 . Note that no inner -face have three common vertices with C .Let f be a k -face with ≤ k ≤ . If f is an internal face, then it does not involve in the dischargingprocedure, and then µ (cid:48) ( f ) = k − ≥ . So we may assume that f is a face in N . If k = 4 and f has exactly one8ommon vertex with C , then it receives from the vertex on the outer cycle C , and then µ (cid:48) ( f ) = 4 − by R6 , R7 and R2 . If k = 4 and f has at least two common vertices with C , then it has exactly one commonedge with C by Lemma 2.7, and then µ (cid:48) ( f ) ≥ − × = by R5 , R6 , R7 and R2 . If k = 5 and f has no common edge with C , then µ (cid:48) ( f ) ≥ − = by R6 , R7 and R2 . If k = 5 and f has acommon edge with C , then it is incident with at most two -vertices and at least one internal vertex, andthen µ (cid:48) ( f ) ≥ − × − × = .Let f be a k -face with ≤ k ≤ . If f is an internal face, then it can only send charges to incident goodvertices, and then µ (cid:48) ( f ) ≥ k − − k × ≥ by R2 . So we may assume that f is a face in N . By R5 , R6 and R7 , the face f receives at least × = from the vertices on the outer cycle C if f has a common edgewith C , and otherwise it receives at least from the vertices on the outer cycle C . Thus, f receives at leasta total of from all the vertices on the outer cycle C . Note that f is incident with at most k − vertices ofdegree two and at least one internal vertex, which implies that µ (cid:48) ( f ) ≥ k − − ( k − × −
13 = 13 ( k − ≥ . (5)Let f be a k -face with k ≥ . By the discharging rules, it is easy to show the following fact. Fact-1. f sends nothing to the + -vertices on the outer cycle C and sends at most to each incident internal -vertex.If f is incident with some -vertices, then f is incident with at least two + -vertices on the outer cycle C and it sends nothing to these vertices, which implies that µ (cid:48) ( f ) ≥ k − − ( k − ×
23 = 13 ( k − > . (6)So we may assume that f is not incident with any -vertex.Let α be the number of bad vertices on f , β be the number of light vertices on f , and let ρ be the numberof internal -vertices on f . It is observed that α + β ≤ ρ . By Lemma 2.5, we can easily show the followingfact on the parameters α and β . Fact-2. If α ≥ , then α + β ≤ ρ ≤ k − .If α ≤ , then µ (cid:48) ( f ) ≥ k − − × − ( k − × = k − ≥ . If α ≥ and k ≥ , then µ (cid:48) ( f ) ≥ k − − ( k − × − × = ( k − ≥ . It remains to assume that α ≥ , k = 9 and f = w w . . . w .Suppose that α = 7 . By Lemma 2.6, the two non-bad vertex divided the bad vertices on f into two parts,one consisting of three consecutive bad vertices and the other consisting of four consecutive bad vertices.Without loss of generality, we may assume that none of w and w is a bad vertex. By Lemma 2.5, w w , w w and w w are incident with -faces. By symmetry, we may further assume that w w and w w areincident with -faces. By Lemma 2.5, w cannot be an internal -vertex. If w is an internal + -vertex or onthe outer cycle C , then f sends nothing to w . If w is an internal -vertex, then w is incident with three + -faces, and it receives nothing from f by R3 . Thus, f sends nothing to w in all cases, which implies that µ (cid:48) ( f ) ≥ − − × − = 0 .If α ≤ , then µ (cid:48) ( f ) ≥ − − × − × − × = 0 . It remains to assume that α = 6 . If there is nolight vertex on f , then µ (cid:48) ( f ) ≥ − − × − × = 0 . So we may assume that there is a light vertex on f . By the definitions of bad vertices and light vertices, a light vertex cannot be adjacent to two bad verticeson f , thus a light vertex must be adjacent to a non-bad vertex on f , which implies that the bad vertices on f are divided into two parts by Lemma 2.6. Without loss of generality, we assume that w is light and w isnon-bad, thus w is bad. Since bad vertex and light vertex cannot be in the same -face, w w is incident9ith a -face and w w is incident with a − -face. If f has a common vertex with C , then f sends nothingto this vertex and µ (cid:48) ( f ) ≥ − − × − × = 0 . So we may assume that f has no common vertex withthe outer cycle C . By Lemma 2.5, w is bad and at least one of { w , w } is not an internal -vertex, andthus it is not bad. Furthermore, w must be a + -vertex. Suppose that w w is incident with a -face. ByLemma 2.5, w w is incident with a -face and none of w and w is an internal -vertex, which implies that w is bad and w w is incident with a -face. By R2 and R3 , f sends to each incident bad vertex, at most to w and sends nothing to w , which implies that µ (cid:48) ( f ) ≥ − − × − − > . In the other case, w w and w w are incident with -faces. Note that one of { w , w } is bad and the other is non-bad, thus w w is incident with a -face. Whenever w or w is non-bad, it receives nothing from f , which implies that µ (cid:48) ( f ) ≥ − − × − × = 0 . Lemma 2.9.
Every vertex has nonnegative final charge.
Proof. If v is a -vertex, then it is on the outer cycle, and then it receives from the incident face in N and from the outer face by R4 , which implies that µ (cid:48) ( v ) = 2 − + = 0 . If v is a -vertex on theouter cycle, then it receives from the outer face and, sends to exactly one − -face in N or sends toeach incident − -face by R5 , which implies that µ (cid:48) ( v ) ≥ − − max (cid:8) , × (cid:9) = 0 . If v is a -vertex onthe outer cycle, then it receives from the outer face and, sends to a special − -face or at most to eachincident face in N by R6 , which implies that µ (cid:48) ( v ) ≥ − − max { , × } = 0 . If v is a + -vertex onthe outer cycle, then it receives from the outer face, and averagely sends at most to each incident face in N , and then µ (cid:48) ( v ) ≥ deg( v ) − − (deg( v ) − × = deg( v ) − ≥ .If v is a bad vertex, then µ (cid:48) ( v ) = 3 − × − = 0 . If v is a light vertex, then µ (cid:48) ( v ) = 3 − × = 0 .If v is a good vertex, then µ (cid:48) ( v ) = 3 − × = 0 . If v is an internal -vertex and incident with twonon-special -faces, then µ (cid:48) ( v ) = 4 − × − × = 0 . If v is an internal -vertex and incident withexactly one non-special -face, then µ (cid:48) ( v ) = 4 − { , × } − = 0 . If v is an internal -vertex butnot incident with any non-special -face, then µ (cid:48) ( v ) = 4 − . If v is an internal + -vertex, then it sendsat most to each incident − -face, which implies that µ (cid:48) ( v ) ≥ deg( v ) − − × (cid:98) deg( v )2 (cid:99) > . Lemma 2.10.
The outer face D has nonnegative charge, and there exists an element having positive finalcharge.By the discharging rules, µ (cid:48) ( D ) ≥ | D | + 4 − | D | = (12 − | D | ) ≥ . The equality holds if and only if | D | = 12 and each vertex on C receives from D . By (4) and R4 – R7 , we may assume that | D | = 12 andeach vertex on C is a − -vertex.Let f be an arbitrary k -face adjacent to D . By the discharging rules, f sends nothing to + -vertices on C and at most to each of the other incident vertex. If f is a + -face, then µ (cid:48) ( f ) ≥ k − − ( k − × > . So wemay assume that every face adjacent to D is an − -face. Since there is no − -faces adjacent to − -faces, eachsuch face is a + -face and each incident internal -vertex is good. By R4 and R2 , f sends to each incident -vertex and at most to each incident internal vertex, which implies that µ (cid:48) ( f ) ≥ k − × − ( k − − = ( k − ≥ , and the equality holds only if f is a -face incident with three -vertices. Let w w . . . w bethe outer cycle, we may assume that w is a -vertex incident with two -faces in N . Thus, w , w , w arethree -vertices and all the other vertices are all -vertices. Now, we have known the structures of G , andthe graph must be as in Fig. 1, a contradiction. Acknowledgments.
This work was supported by the National Natural Science Foundation of China andpartially supported by the Fundamental Research Funds for Universities in Henan (YQPY20140051).10 eferences [1] O. V. Borodin, Colorings of plane graphs: A survey, Discrete Math. 313 (4) (2013) 517–539.[2] L. Chen, R. Liu, G. Yu, R. Zhao and X. Zhou, DP-4-colorability of two classes of planar graphs, DiscreteMath. 342 (11) (2019) 2984–2993.[3] Z. Dvořák and L. Postle, Correspondence coloring and its application to list-coloring planar graphswithout cycles of lengths 4 to 8, J. Combin. Theory Ser. B 129 (2018) 38–54.[4] H. Grötzsch, Zur Theorie der diskreten Gebilde. VII. Ein Dreifarbensatz für dreikreisfreie Netze auf derKugel, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math.-Nat. Reihe 8 (1959) 109–120.[5] R. Li and T. Wang, DP-4-coloring of planar graphs with some restrictions on cycles, arXiv:1909.08511, https://arxiv.org/abs/1909.08511 .[6] R. Li and T. Wang, Variable degeneracy on toroidal graphs, arXiv:1907.07141, https://arxiv.org/abs/1907.07141 .[7] R. Liu and X. Li, Every planar graph without adjacent cycles of length at most 8 is 3-choosable, EuropeanJ. Combin. 82 (2019) 102995, 10.[8] R. Liu, X. Li, K. Nakprasit, P. Sittitrai and G. Yu, DP-4-colorability of planar graphs without adjacentcycles of given length, Discrete Appl. Math. (2020) https://doi.org/10.1016/j.dam.2019.09.012 .[9] R. Liu, S. Loeb, M. Rolek, Y. Yin and G. Yu, DP-3-coloring of planar graphs without 4, 9-cycles andcycles of two lengths from { , , } , Graphs Combin. 35 (3) (2019) 695–705.[10] R. Liu, S. Loeb, Y. Yin and G. Yu, DP-3-coloring of some planar graphs, Discrete Math. 342 (1) (2019)178–189.[11] X. Luo, M. Chen and W. Wang, On 3-colorable planar graphs without cycles of four lengths, Inform.Process. Lett. 103 (4) (2007) 150–156.[12] C. Thomassen, Every planar graph is -choosable, J. Combin. Theory Ser. B 62 (1) (1994) 180–181.[13] C. Thomassen, -list-coloring planar graphs of girth5