A star edge coloring of a graph $G$ is a proper edge coloring with no 2-colored path or cycle of length four. The star edge coloring problem is to find an edge coloring of a given graph $G$ with minimum number $k$ of colors such that $G$ admits a star edge coloring with $k$ colors. This problem is known to be NP-complete. In this paper, for a bounded treewidth graph with given maximum degree, we show that it can be solved in polynomial time.
We conjecture that any graph $G$ with treewidth~$k$ and maximum degree $Δ(G)\geq k + \sqrt{k}$ satisfies $χ'(G)=Δ(G)$. In support of the conjecture we prove its fractional version. We also show that any graph $G$ with treewidth~$k\geq 4$ and maximum degree $2k-1$ satisfies $χ'(G)=Δ(G)$, improving an old result of Vizing.
The clustered chromatic number of a graph class is the minimum integer $t$ such that for some $C$ the vertices of every graph in the class can be colored in $t$ colors so that every monochromatic component has size at most $C$. We show that the clustered chromatic number of the class of graphs embeddable on a given surface is four, proving the conjecture of Esperet and Ochem. Additionally, we study the list version of the concept and characterize the minor-closed classes of graphs of bounded treewidth with given clustered list chromatic number. We further strengthen the above results to solve …
A long-standing conjecture asserts that there exists a constant $c>0$ such that every graph of order $n$ without isolated vertices contains an induced subgraph of order at least $cn$ with all degrees odd. Scott (1992) proved that every graph $G$ has an induced subgraph of order at least $|V(G)|/(2χ(G))$ with all degrees odd, where $χ(G)$ is the chromatic number of $G$, this implies the conjecture for graphs with { bounded} chromatic number. But the factor $1/(2χ(G))$ seems to be not best possible, for example, Radcliffe and Scott (1995) proved $c=\frac 23$ for trees, Berman, Wang and Wargo (19…
The dichromatic number of a digraph $G$ is the smallest integer $χ_a(G)$ such that the vertex set of $G$ can be partitioned into $χ_a(G)$ sets, each of which induces an acyclic subdigraph. This is a generalization of the classic chromatic number of graphs. Here, we investigate the dichromatic number of the cartesian, direct, strong and lexicographic products, giving generalizations of some classic results on the chromatic number of products. More specifically, we prove that the following inequalities, known to hold for the chromatic number of graphs, still hold for the dichromatic number of di…
We consider the parameterised complexity of several list problems on graphs, with parameter treewidth or pathwidth. In particular, we show that List Edge Chromatic Number and List Total Chromatic Number are fixed parameter tractable, parameterised by treewidth, whereas List Hamilton Path is W[1]-hard, even parameterised by pathwidth. These results resolve two open questions of Fellows, Fomin, Lokshtanov, Rosamond, Saurabh, Szeider and Thomassen (2011).
Signed list edge coloring in graphs of bounded treewidth
Yan’an University（延安大学）；School of Mathematics and Statistics, Northwestern Polytechnical University（西北工业大学数学与统计学院）；School of Mathematics and Statistics & Key Laboratory of Analytical Mathematics and Applications (Ministry of Education), Fujian Normal University（福建师范大学数学与统计学院与分析数学及应用重点实验室（教育部））；School of Marine Science and Technology, Northwestern Polytechnical University（西北工业大学海洋科学与技术学院）
The distinguishing index of a graph $G$, denoted by $D'(G)$, is the least number of labels in an edge labeling of $G$ not preserved by any non-trivial automorphism. The distinguishing chromatic index $χ'_D (G)$ of a graph $G$ is the least number $d$ such that $G$ has a proper edge labeling with $d$ labels that is preserved only by the identity automorphism of $G$. In this paper we compute the distinguishing chromatic index for some specific graphs. Also we study the distinguishing chromatic index of corona product and join of two graphs.
We consider class of integral sum graphs $H^{-i,s}_{m,j}$ subject to the conditions $-i<0<s$, $1\leq m < i$ and $1\leq j < s$ for all $i,s, m,j\in \mathbb{N}$. We apply edge-sum coloring and edge coloring on $H^{-i,s}_{m,j}$. Since the graphs fully depend on $i$ and $s$, therefore it is not easy to derive the theoretical as well as numerical results for all values of $i$ and $s$. Here, we derive the general formula for computing the minimum number of independent color classes. We compute the edge chromatic as well as edge-sum chromatic number of $H^{-i,s}_{m,j}$ corresponding to different valu…
Fong et al. (The game chromatic index of some trees with maximum degree four and adjacent degree-four vertices, J. Comb Optim 36 (2018) 1-12) proved that the game chromatic index of any tree $T$ of maximum degree 4 whose degree-four vertices induce a forest of paths of length $l$ less than 2 is at most 5. In this paper, we show that the bound 5 is also valid for $l\leq 2$. This partially solves the problem of characterization of the trees whose game chromatic index exceeds the maximum degree by at most 1, which was proposed by Cai and Zhu (Game chromatic index of $k$-degenerate graphs, J. Grap…
A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every path and cycle of length four in $G$ uses at least three different colors. The star chromatic index of a graph $G$, is the smallest integer $k$ for which $G$ admits a star edge coloring with $k$ colors. In this paper, we present a polynomial time algorithm that finds an optimum star edge coloring for every tree. We also provide some tight bounds on the star chromatic index of trees with diameter at most four, and using these bounds we find a formula for the star chromatic index of certain families of trees.
The planar graph product structure theorem of Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] states that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. This result has been the key tool to resolve important open problems regarding queue layouts, nonrepetitive colourings, centered colourings, and adjacency labelling schemes. In this paper, we extend this line of research by utilizing shallow minors to prove analogous product structure theorems for several beyond planar graph classes. The key observation that drives our work …
A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every path and cycle of length four in $G$ uses at least three different colors. The star chromatic index of $G$, is the smallest integer $k$ for which $G$ admits a star edge coloring with $k$ colors. In this paper, we obtain tight upper bound $\left\lfloor\frac{3Δ}{2}\right\rfloor+2$ for the star chromatic index of every Halin graph, that proves the conjecture of Dvo{ř}{á}k et al. (J Graph Theory, 72 (2013), 313--326) for cubic Halin graphs.
The $\!{}\bmod k$ chromatic index of a graph $G$ is the minimum number of colors needed to color the edges of $G$ in a way that the subgraph spanned by the edges of each color has all degrees congruent to $1\!\!\pmod k$. Recently, the authors proved that the $\!{}\bmod k$ chromatic index of every graph is at most $198k-101$, improving, for large $k$, a result of Scott [Discrete Math. 175, 1-3 (1997), 289-291]. Here we study the $\!{}\bmod k$ chromatic index of random graphs. We prove that for every integer $k\geq2$, there is $C_k>0$ such that if $p\geq C_kn^{-1}\log{n}$ and $n(1-p) \rightarrow…
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].
A strong $k$-edge-coloring of a graph G is an edge-coloring with $k$ colors in which every color class is an induced matching. The strong chromatic index of $G$, denoted by $χ'_{s}(G)$, is the minimum $k$ for which $G$ has a strong $k$-edge-coloring. In 1985, Erdős and Nešetřil conjectured that $χ'_{s}(G)\leq\frac{5}{4}Δ(G)^2$, where $Δ(G)$ is the maximum degree of $G$. When $G$ is a graph with maximum degree at most 3, the conjecture was verified independently by Andersen and Horák, Qing, and Trotter. In this paper, we consider the list version of strong edge-coloring. In particular, we show …
It is conjectured that every Steiner triple system of order $v \neq 7$ has chromatic index at most $(v+3)/2$ when $v \equiv 3 \pmod{6}$ and at most $(v+5)/2$ when $v \equiv 1 \pmod{6}$. Herein, we construct a Steiner triple system of order $v$ with chromatic index at least $(v+3)/2$ for each integer $v \equiv 3 \pmod{6}$ such that $v \geq 15$, with four possible exceptions. We further show that the maximum number of disjoint parallel classes in the systems constructed is sublinear in $v$. Finally, we establish for each order $v \equiv 15 \pmod{18}$ that there are at least $v^{v^2(1/6+o(1))}$ n…
A vertex coloring of a given graph $G$ is conflict-free if the closed neighborhood of every vertex contains a unique color (i.e. a color appearing only once in the neighborhood). The minimum number of colors in such a coloring is the conflict-free chromatic number of $G$, denoted $χ_{CF}(G)$. What is the maximum possible conflict-free chromatic number of a graph with a given maximum degree $Δ$? Trivially, $χ_{CF}(G)\leq χ(G)\leq Δ+1$, but it is far from optimal - due to results of Glebov, Szabó and Tardos, and of Bhyravarapu, Kalyanasundaram and Mathew, the answer in known to be $Θ\left(\ln^2Δ…
Using an exhaustive search on cubic graphs of order 16, we find a unique cospectral pair with different chromatic indexes. This example indicates that the chromatic index of a regular graph is not characterized by its spectrum, which answers a question recently posed in [O. Etesami, W. H. Haemers, On NP-hard graph properties characterized by the spectrum, Discrete Appl. Math., 285(2020)526-529]. We prove that any orthogonal matrix representing the similarity between the two adjacency matrices of the cospectral pair cannot be rational. This implies that the cospectral pair cannot be obtained us…
Given a graph $G$ possibly with multiple edges but no loops, denote by $Δ$ the {\it maximum degree}, $μ$ the {\it multiplicity}, $χ'$ the {\it chromatic index} and $χ_f'$ the {\it fractional chromatic index} of $G$, respectively. It is known that $Δ\le χ_f' \le χ' \le Δ+ μ$, where the upper bound is a classic result of Vizing. While deciding the exact value of $χ'$ is a classic NP-complete problem, the computing of $χ_f'$ is in polynomial time. In fact, it is shown that if $χ_f' > Δ$ then $χ_f'= \max \frac{|E(H)|}{\lfloor |V(H)|/2\rfloor}$, where the maximality is over all induced subgraphs $H…
We show that for a simple graph $G$, $c'(G)\leqΔ(G)+2$ where $c'(G)$ is the choice index (or edge-list chromatic number) of $G$, and $Δ(G)$ is the maximum degree of $G$. As a simple corollary of this result, we show that the total chromatic number $χ_T(G)$ of a simple graph satisfies the inequality $χ_T(G)\leq\ Δ(G)+4$ and the total choice number $c_T(G)$ also satisfies this inequality. We also relate these bounds to the Hall index and the Hall condition index of a simple graph, and to the total Hall number and the total Hall condition number of a simple graph.
The acyclic chromatic index of a graph $G$ is the least number of colors needed to properly color its edges so that none of its cycles is bichromatic. In this work, we show that $2Δ-1$ colors are sufficient to produce such a coloring, where $Δ$ is the maximum degree of the graph. In contrast with most extant randomized algorithmic approaches to the chromatic index, where the algorithms presuppose enough colors to guarantee properness deterministically and use randomness only to deal with the bichromatic cycles, our randomized, Moser-type algorithm produces a not necessarily proper random color…
Let $G$ be a simple graph with maximum degree $Δ(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>Δ(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ on $n$ vertices with $Δ(G)>n/3$ has chromatic index $Δ(G)$ if and only if $G$ contains no overfull subgraph. Glock, Kühn 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, Kühn and Osthus is affirmative.
Let $G$ be a graph with maximum degree $Δ$ and without isolated vertices. An edge colouring $c$ of $G$ is conflict-free if the closed neighbourhood of every edge includes a uniquely coloured element. The least number of colours admitting such $c$ is the conflict-free chromatic index of $G$, denoted by $χ'_{CF}(G)$. In "Conflict-free chromatic number versus conflict-free chromatic index" [J. Graph Theory, 2022; 99: 349--358] it was recently proved by means of the probabilistic method that $χ'_{CF}(G)\leq C_1\log_2Δ+C_2$, where $C_1>337$ and $C_2$ are constants, whereas there are families of gra…
A strong edge-coloring of a graph $G=(V,E)$ is a partition of its edge set $E$ into induced matchings. We study bipartite graphs with one part having maximum degree at most $3$ and the other part having maximum degree $Δ$. We show that every such graph has a strong edge-coloring using at most $3 Δ$ colors. Our result confirms a conjecture of Brualdi and Quinn Massey ~\cite{[BQ]} for this class of bipartite graphs.
The $χ$-stability index ${\rm es}_χ(G)$ 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 ($k\le 5$) graphs $G$ with ${\rm es}_χ(G) = 1$ does not extend to $k\ge 6$. Graphs $G$ with $χ(G)=3$ for which ${\rm es}_χ(G)+{\rm es}_χ(\overline{G}) = 2$ holds are characterized. Necessary conditions on graphs $G$ which attain a known uppe…
Give a digraph $D=(V(D),A(D))$, let $\partial^+_D(v)=\{vw|w\in N^+_D(v)\}$ and $\partial^-_D(v)=\{uv|u\in N^-_D(v)\}$ be semi-cuts of $v$. A mapping $φ:A(D)\rightarrow [k]$ is called a weak-odd $k$-edge coloring of $D$ if it satisfies the condition: for each $v\in V(D)$, there is at least one color with an odd number of occurrences on each non-empty semi-cut of $v$. We call the minimum integer $k$ the weak-odd chromatic index of $D$. When limit to 2 colors, use $def(D)$ to denote the defect of $D$, the minimum number of vertices in $D$ at which the above condition is not satisfied. In this pap…
A star edge coloring of a graph $G$ is a proper edge coloring of $G$ without bichromatic paths or cycles of length four. The it star chromatic index, $χ_{st}^{'} (G ),$ of $G$ is the minimum number $k$ for which $G$ has a star edge coloring by $k$ colors. In \cite{LB}, L. Bezegov$\acute{a}$ et al. conjectured that $χ_{st}^{'} (G )\leq \lfloor\frac{3Δ}{2}\rfloor+1$ when $G$ is an outerplanar graph with maximum degree $Δ\geq 3.$ In this paper we obtained that $χ_{st}^{'}(G) \leq Δ+6$ when $G$ is an 2-connected outerplanar graph with diameter 2 or 3. If $G$ is an 2-connected outerplanar graph wit…
We study the class of simple graphs $\mathcal{G}^*$ for which every pair of distinct odd cycles intersect in at most one edge. We give a structural characterization of the graphs in $\mathcal{G}^*$ and prove that every $G \in \mathcal{G}^*$ satisfies the list-edge-coloring conjecture. When $Δ(G) \geq 4$, we in fact prove a stronger result about kernel-perfect orientations in $L(G)$ which implies that $G$ is $(mΔ(G):m)$-edge-choosable and $Δ(G)$-edge-paintable for every $m \geq 1$.
Let $χ'_k(G)$ denote the minimum number of colors needed to color the edges of a graph $G$ in a way that the subgraph spanned by the edges of each color has all degrees congruent to $1 \pmod k$. Scott [{\em Discrete Math. 175}, 1-3 (1997), 289--291] proved that $χ'_k(G)\leq5k^2\log k$, and thus settled a question of Pyber [{\em Sets, graphs and numbers} (1992), pp. 583--610], who had asked whether $χ_k'(G)$ can be bounded solely as a function of $k$. We prove that $χ'_k(G)=O(k)$, answering affirmatively a question of Scott.
