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Star Chromatic Index of Subcubic Multigraphs Received: 15 January 2017 Revised: 27 October 2017 Accepted: 20 November 2017 DOI: 10.1002/jgt.22230 ARTICLE Star chromatic index of subcubic multigraphs Hui Lei1 Yongtang Shi1 Zi-Xia Song2 1 Center for Combinatorics and LPMC, Nankai Abstract University, Tianjin 300071, China The star chromatic index of a multigraph , denoted ′(), 2Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA is the minimum number of colors needed to properly color Correspondence the edges of such that no path or cycle of length four Zi-Xia Song, Department of Math- is bicolored. A multigraph is star -edge-colorable if ematics, University of Central ′ ≤ Florida, Orlando, FL 32816, USA. ( ) . Dvořák, Mohar, and Šámal [Star chromatic Email: [email protected] index, J. Graph Theory 72 (2013), 313–326] proved that Contract grant sponsor: National Natural Sci- every subcubic multigraph is star 7-edge-colorable. They ence Foundation of China and Natural Science conjectured in the same article that every subcubic multi- Foundation of Tianjin; contract grant number: 17JCQNJC00300. graph should be star 6-edge-colorable. In this article, we first prove that it is NP-complete to determine whether ′ () ≤ 3 for an arbitrary graph . This answers a ques- tion of Mohar. We then establish some structure results on ′ subcubic multigraphs with () ≤ 2 such that () > ′ but ( − ) ≤ for any ∈ (), where ∈{5, 6}. We finally apply the structure results, along with a sim- ple discharging method, to prove that every subcubic multi- graph is star 6-edge-colorable if () < 5∕2, and star 5-edge-colorable if () < 24∕11, respectively, where () is the maximum average degree of a multigraph . This partially confirms the conjecture of Dvořák, Mohar, and Šámal. KEYWORDS maximum average degree, star edge-coloring, subcubic multigraphs 1 INTRODUCTION All multigraphs in this article are finite and loopless; and all graphs are finite and without loops or multiple edges. Given a multigraph ,let ∶ () → [] be a proper edge-coloring of , where ≥ 1 is an integer and []∶={1, 2, … ,}. We say that is a star -edge-coloring of if no path or J Graph Theory. 2017;1–11. wileyonlinelibrary.com/journal/jgt © 2017 Wiley Periodicals, Inc. 1 2 LEI ET AL. cycle of length four in is bicolored under the coloring ; and is star -edge-colorable if admits ′ astar-edge-coloring. The star chromatic index of , denoted by (), is the smallest integer such that is star -edge-colorable. The chromatic index of is denoted by ′(). As pointed out in [7], the definition of star edge-coloring of a graph is equivalent to the star vertex-coloring of its line graph (). Star edge-coloring of a graph was initiated by Liu and Deng [10], motivated by the vertex version (see [1,5,6,9,11]). Given a multigraph ,weuse|| to denote the number of vertices, () the number of edges, () the minimum degree, and Δ() the maximum degree of , respectively. For any ∈ ( ),let ( ) and ( ) denote⋃ the degree and neighborhood of in , respectively. For , ⊆ any subsets ( ),let ( )∶= ∈ ( ), and let ∖ ∶= − .If ={ },wesimply write ∖ instead of ∖.Weuse and to denote the complete graph and the path on vertices, respectively. It is well-known [13] that the chromatic index of a graph with maximum degree Δ is either Δ or Δ+1. However, it is NP-complete [8] to determine whether the chromatic index of an arbitrary graph with maximum degree Δ is Δ or Δ+1. The problem remains NP-complete even for cubic graphs. A multigraph is subcubic if the maximum degree of is at most three. Mohar (private communica- ′ tion with the second author) proposed that it is NP-complete to determine whether () ≤ 3 for an arbitrary graph . We first answer this question in the positive. ′ Theorem 1.1. It is NP-complete to determine whether () ≤ 3 for an arbitrary graph . We prove Theorem 1.1 in Section 2. Theorem 1.2 below is a result of Dvořák et al. [7], which gives an upper bound and a lower bound for complete graphs. Theorem 1.2 ( [7]). The star chromatic index of the complete graph satisfies √ √ 2 2(1+(1)) log ′ 2 2(1 + (1)) ≤ () ≤ . (log )1∕4 ′ 1+ In particular, for every >0, there exists a constant such that () ≤ for every integer ≥ 1. ′ The true order of magnitude of ( ) is still unknown. From Theorem 1.2, an upper bound√ in ′ (1) log Δ terms of the maximum degree for general graphs is also derived in [7], i.e. () ≤ Δ ⋅ 2 for any graph with maximum degree Δ. In the same article, Dvořák et al. [7] also considered the star chromatic index of subcubic multigraphs. To state their result, we need to introduce one notation. A graph covers a graph if there is a mapping ∶ () → () such that for any ∈ (), ()()∈(), and for any ∈ (), is a bijection between () and (()).Theyproved the following. Theorem 1.3 ( [7]). Let be a multigraph. ′ (a) If is subcubic, then () ≤ 7. ′ (b) If is cubic and has no multiple edges, then () ≥ 4 and the equality holds if and only if covers the graph of 3-cube. ′ As observed in [7], ( 3,3)=6and the Heawood graph is star 6-edge-colorable. No subcubic multigraphs with star chromatic index seven are known. Dvořák et al. [7] proposed the following con- jecture. ′ Conjecture 1.4. Let be a subcubic multigraph. Then () ≤ 6. LEI ET AL. 3 As far as we know, not much progress has been made yet toward Conjecture 1.4. It was recently shown in [2] that every subcubic outerplanar graph is star 5-edge-colorable. A tight upper bound for trees was also obtained in [2]. We summarize the main results in [2] as follows. Theorem 1.5 ( [2]). Let be an outerplanar graph. Then ⌊ ⌋ 3Δ() (a) ′() ≤ if is a tree. Moreover, the bound is tight. 2 (b) ′() ≤ 5 if Δ() ≤ 3. ⌊ ⌋ 3Δ() (c) ′() ≤ +12if Δ() ≥ 4. 2 The maximum average degree of a multigraph , denoted (), is defined as the maximum of 2()∕|| taken over all the subgraphs of . We want to point out here that there is an error in the proof of Theorem 2.3 in a recent published article by Pradeep and Vijayalakshmi [Star chromatic index of subcubic graphs, Electronic Notes in Discrete Mathematics 53 (2016), 155–164]. Theorem ′ 2.3 in [12] claims that if is a subcubic graph with () < 11∕5, then () ≤ 5. The error in the proof of Theorem 2.3 arises from ambiguity in the statement of Claim 3 in their article. From its proof given in [12] (on page 158), Claim 3 should be stated as “ does not contain a path , where either all of , , are 2-vertices or all of , , are light 3-vertices.” This new statement of Claim 3 does not imply that “a 2-vertex must be adjacent to a heavy 3-vertex” in Case 2 of the proof of Theorem 2.3 (on page 162). It seems nontrivial to fix this error in their proof. If Claim 3 in their article is true, using the technique we developed in the proof of Theorem 1.6(b), one can obtain a stronger result that every subcubic multigraph with () < 7∕3 is star 5-edge-colorable. In this article, we prove two main results, namely Theorem 1.1 mentioned above and Theorem 1.6 below. Theorem 1.6. Let be a subcubic multigraph. ′ (a) If () < 2, then () ≤ 4 and the bound is tight. ′ (b) If () < 24∕11, then () ≤ 5. ′ (c) If () < 5∕2, then () ≤ 6. The rest of this article is organized as follows. We prove Theorem 1.1 in Section 2. Before we prove Theorem 1.6 in Section 4, we establish in Section 3 some structure results on subcubic multigraphs ′ ′ with () ≤ 2 such that () >and ( − ) ≤ for any ∈ (), where ∈{5, 6}. We believe that our structure results can be used to solve Conjecture 1.4. 2 PROOF OF THEOREM 1.1 First let us denote by SEC the problem stated in Theorem 1.1, and we denote by 3EC the following well-known NP-complete problem of Holyer [8]: Given a cubic graph ,is 3-edge-colorable? Proof of Theorem 1.1. Clearly, SEC is in the class NP. We shall reduce 3EC to SEC. Let be an instance of 3EC. We construct a graph from by replacing each edge = ∈ () with a copy of graph , identifying with and with , where is depicted in Figure 1. The size of is clearly polynomial in the size of , and Δ()=3. 4 LEI ET AL. FIGURE 1 Graph ′ ′ ′ It suffices to show that () ≤ 3 if and only if () ≤ 3. Assume that () ≤ 3.Let ∶ () → {1, 2, 3} be a proper 3-edge-coloring of .Let∗ be an edge coloring of obtained from ∗ ∗ ∗ ∗ as follows: for each edge = ∈ ( ),let ( 1)= ( 3 4)= ( 6 )= ( ), ( 1 2)= ∗ ∗ ∗ ∗ ∗ ( 3 7)= ( 4 8)= ( 5 6)= ( )+1, and ( 2 3)= ( 4 5)= ( )+2, where all colors here and henceforth are done modulo 3. Notice that ∗ is a proper 3-edge-coloring of . Furthermore, it can be easily checked that has no bicolored path or cycle of length four under the coloring ∗. Thus ∗ ′ is a star 3-edge-coloring of and so () ≤ 3. ′ ∗ Conversely, assume that () ≤ 3.Let ∶ () → {1, 2, 3} be a star 3-edge-coloring of .Let ∗ ∗ be an edge-coloring of obtained from by letting ( )= ( 1) for any = ∈ ( ). Clearly, ∗ ∗ is a proper 3-edge-coloring of if for any edge = in , ( 1)= ( 6 ). We prove this next. Let = be an edge of . We consider the following two cases. Case 1: ∗ ∗ ( 3 7)= ( 4 8). ∗ ∗ , , In this case, let ( 3 7)= ( 4 8)= , where ∈{1 2 3}. We may further assume ∗ ∗ ∗ , , , that ( 3 4)= and ( 2 3)= ( 4 5)= , where { }={1 2 3}∖ .
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