Acyclic edge coloring conjecture is true on planar graphs without intersecting triangles☆
Introduction
Let G be a simple graph with vertex set and edge set . For an integer , a (proper) edge k-coloring is a mapping such that any two adjacent edges receive different colors. (We drop “proper” in the sequel.) G is edge k-colorable if G has an edge k-coloring. The chromatic index of G is the smallest integer k such that G is edge k-colorable. An edge k-coloring c of G is called acyclic if there are no bichromatic cycles in G, i.e., the subgraph of G induced by any two colors is a forest. The acyclic chromatic index of G, denoted by , is the smallest integer k such that G is acyclically edge k-colorable.
Recall that a connected graph has a block-cut tree representation in which each block is a 2-connected component and two blocks overlap at most one cut-vertex. Since we may swap any two colors inside a block, if necessary, in the context of edge coloring or acyclic edge coloring we assume w.l.o.g. that G is 2-connected. In the sequel, only simple and 2-connected graphs are considered in this paper.
Let (Δ for short and reserved) denote the maximum degree of the graph G. One sees that . Note that by Vizing's theorem [18] and that . Fiamčik [7] and Alon, Sudakov and Zaks [2] independently made the following acyclic edge coloring conjecture (AECC):
Conjecture 1 (AECC) For any graph G, .
For an arbitrary graph G, the following milestones have been achieved: Alon, McDiarmid and Reed [1] proved that by a probabilistic argument. The upper bound was improved to 16Δ [11], to [12], to [6], to by Giotis et al. [8] using the Lovász local lemma, and most recently in 2020 to by Kirousis and Livieratos [10] using a Moser-type randomized algorithm. On the other hand, the AECC has been confirmed true for graphs with [17], [3], [4], [16], [21].
When G is planar, i.e., G can be drawn in the two-dimensional plane so that its edges intersect only at their ending vertices, Basavaraju et al. [5] showed that . The upper bound was improved to by Wang, Shu and Wang [23] and to by Wang and Zhang [20]. The AECC has been confirmed true for planar graphs without i-cycles for each in [15], [22], [14], [24], respectively.
A triangle is synonymous with a 3-cycle. We say that two triangles are adjacent if they share a common edge, and are intersecting if they share at least a common vertex. Recall that the truth of the AECC for planar graphs without triangles has been verified in [15]. When a planar graph G contains triangles but no intersecting triangles, Hou, Roussel and Wu [9] proved the upper bound , and Wang and Zhang [19] improved it to . This paper focuses on planar graphs without intersecting triangles too, and we completely resolve the AECC by showing the following main theorem.
Theorem 1 The AECC is true for planar graphs without intersecting triangles.
The rest of the paper is organized as follows. In Section 2, we characterize six groups of local structures (also called configurations), and by discharging methods we prove that any planar graph without intersecting triangles must contain at least one of these local structures. Incorporating a known property of edge colorings and bichromatic cycles (Lemma 9), in Section 3 we prove by induction on the number of edges that the graph admits an acyclic edge -coloring. We conclude the paper in Section 4.
Section snippets
The six groups of local structures
Recall that we consider only simple 2-connected planar graphs.
Given a graph G, let denote the degree of the vertex v in G. A vertex of degree k (at least k, at most k, respectively) is called a k-vertex (-vertex, -vertex, respectively); and it is called a k-neighbor (-neighbor, -neighbor, respectively) of any adjacent vertex. Let (, , respectively) denote the number of k-vertices (-vertices, -vertices, respectively) adjacent to v in G.
Theorem 2 Let G be a simple 2
Acyclic edge coloring
In this section, we show how to derive an acyclic edge coloring, by an induction on and by recoloring certain edges in each specified local structure. Recall that G is a simple 2-connected planar graph without intersecting triangles. The following lemma gives the starting point.
Lemma 8 ([17], [3], [4], [16], [21]) If , then , and an acyclic edge -coloring can be obtained in polynomial time.
Given a partial acyclic edge k-coloring of the graph G using the color set
Conclusions
Proving the acyclic edge coloring conjecture (AECC) on general graphs has a long way to go, but on planar graphs it is very close with recent efforts. In particular, the AECC has been confirmed true for planar graphs without i-cycles, for each . In this paper, we dealt with the planar graphs containing 3-cycles but no intersecting 3-cycles. Prior to our work, their acyclic chromatic index was shown to be at most , and in this work we closed the AECC by confirming it affirmatively.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors are very grateful to the anonymous reviewers for their helpful comments and suggestions.
QS is supported by the ZJNSF Grants LY20F030007 and LQ15A010010 and the NSFC Grant 11601111. YC and AZ are supported by the NSFC Grants 11971139 and 11771114, and by the China Scholarship Council Grants 201508330054 and 201908330090, respectively. GL is supported by the NSERC Canada. EM is supported by the KAKENHI Grants JP21K11755 and JP17K00016 and the JST CREST JPMJR1402.
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