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Register a new userA new bound on the acyclic edge chromatic index
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Bernardo Nunes Borges de Lima
Publication date
2019
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English
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In this note we obtain a new bound for the acyclic edge chromatic number $a(G)$ of a graph $G$ with maximum degree $D$ proving that $a(G)leq 3.569(D-1)$. To get this result we revisit and slightly modify the method described in [Giotis, Kirousis, Psaromiligkos and Thilikos, Theoretical Computer Science, 66: 40-50, 2017].
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An adjacent vertex distinguishing coloring of a graph G is a proper edge coloring of G such that any pair of adjacent vertices are incident with distinct sets of colors. The minimum number of colors needed for an adjacent vertex distinguishing coloring of G is denoted by $chi_a(G)$. In this paper, we prove that $chi_a(G)$ <= 5($Delta+2$)/2 for any graph G having maximum degree $Delta$ and no isolated edges. This improves a result in [S. Akbari, H. Bidkhori, N. Nosrati, r-Strong edge colorings of graphs, Discrete Math. 306 (2006), 3005-3010], which states that $chi_a(G)$ <= 3$Delta$ for any graph G without isolated edges.
The strong chromatic index of a graph $G$, denoted $chi_s(G)$, is the least number of colors needed to edge-color $G$ so that edges at distance at most two receive distinct colors. The strong list chromatic index, denoted $chi_{s,ell}(G)$, is the least integer $k$ such that if arbitrary lists of size $k$ are assigned to each edge then $G$ can be edge-colored from those lists where edges at distance at most two receive distinct colors. We use the discharging method, the Combinatorial Nullstellensatz, and computation to show that if $G$ is a subcubic planar graph with $operatorname{girth}(G) geq 41$ then $chi_{s,ell}(G) leq 5$, answering a question of Borodin and Ivanova [Precise upper bound for the strong edge chromatic number of sparse planar graphs, Discuss. Math. Graph Theory, 33(4), (2014) 759--770]. We further show that if $G$ is a subcubic planar graph and $operatorname{girth}(G) geq 30$, then $chi_s(G) leq 5$, improving a bound from the same paper. Finally, if $G$ is a planar graph with maximum degree at most four and $operatorname{girth}(G) geq 28$, then $chi_s(G) leq 7$, improving a more general bound of Wang and Zhao from [Odd graphs and its application on the strong edge coloring, arXiv:1412.8358] in this case.
The $chi$-stability index ${rm es}_{chi}(G)$ of a graph $G$ is the minimum number of its edges whose removal results in a graph with the chromatic number smaller than that of $G$. In this paper three open problems from [European J. Combin. 84 (2020) 103042] are considered. Examples are constructed which demonstrate that a known characterization of $k$-regular ($kle 5$) graphs $G$ with ${rm es}_{chi}(G) = 1$ does not extend to $kge 6$. Graphs $G$ with $chi(G)=3$ for which ${rm es}_{chi}(G)+{rm es}_{chi}(overline{G}) = 2$ holds are characterized. Necessary conditions on graphs $G$ which attain a known upper bound on ${rm es}_{chi}(G)$ in terms of the order and the chromatic number of $G$ are derived. The conditions are proved to be sufficient when $nequiv 2 pmod 3$ and $chi(G)=3$.
A strong edge colouring of a graph is an assignment of colours to the edges of the graph such that for every colour, the set of edges that are given that colour form an induced matching in the graph. The strong chromatic index of a graph $G$, denoted by $chi_s(G)$, is the minimum number of colours needed in any strong edge colouring of $G$. A graph is said to be emph{chordless} if there is no cycle in the graph that has a chord. Faudree, Gyarfas, Schelp and Tuza~[The Strong Chromatic Index of Graphs, Ars Combin., 29B (1990), pp.~205--211] considered a particular subclass of chordless graphs, namely the class of graphs in which all the cycle lengths are multiples of four, and asked whether the strong chromatic index of these graphs can be bounded by a linear function of the maximum degree. Chang and Narayanan~[Strong Chromatic Index of 2-degenerate Graphs, J. Graph Theory, 73(2) (2013), pp.~119--126] answered this question in the affirmative by proving that if $G$ is a chordless graph with maximum degree $Delta$, then $chi_s(G) leq 8Delta -6$. We improve this result by showing that for every chordless graph $G$ with maximum degree $Delta$, $chi_s(G)leq 3Delta$. This bound is tight up to to an additive constant.
Let $G$ be a simple graph with maximum degree $Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>Delta(G)lfloor |V(H)|/2 rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ on $n$ vertices with $Delta(G)>n/3$ has chromatic index $Delta(G)$ if and only if $G$ contains no overfull subgraph. Glock, K{u}hn and Osthus in 2016 showed that the conjecture is true for dense quasirandom graphs with even order, and they conjectured that the same should hold for such graphs with odd order. In this paper, we show that the conjecture of Glock, K{u}hn and Osthus is affirmative.
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