A strong edge coloring is a proper edge coloring in which every color class is an induced matching; the least number of colors is the strong chromatic index $\chi'_s(G)$. Lin and Lin proved that every claw-free subcubic graph other than the triangular prism satisfies $\chi'_s(G) \le 7$, with all their tight examples containing diamonds. Kardos (Problem 4.1 of the open-problem collection of the 33rd Workshop on Cycles and Colourings) asked whether every diamond-free claw-free cubic graph is strongly 6-edge-colorable, equivalently whether $\chi'_s(T(G))=6$ for every cubic graph $G$, where $T(G)$ is the truncation of $G$. We exhibit an explicit connected, simple, diamond-free, claw-free cubic graph $H$ on 18 vertices with $\chi'_s(H)=7$, and show that it has the fewest vertices possible for such a non-prism example. Hence the first formulation, over simple cubic graphs, is false even after excluding the prism; and since $H$ is the truncation of a cubic multigraph with parallel edges, the two formulations are not equivalent unless"cubic graph"is allowed to mean loopless multigraph, under which reading the problem is answered negatively. The narrower version restricted to truncations of simple base graphs remains open.
We prove that there is an absolute constant $c>0$ such that every graph of chromatic number at least $r$ and at most $cr^3\log^2 r$ edges contains at least $\binom r3$ triangles. The proof has three ingredients. First, a sparse-core argument based on a triangle-sensitive coloring estimate of Harris extracts, from any c...
A strong edge colouring of a graph $G$ is an edge colouring in which every colour class is an induced matching. The minimum number of colours is the strong chromatic index $\chi'_s(G)$. If each edge $e$ is assigned a list $L'(e)$ and its colour must belong to $L'(e)$, the corresponding parameter is the strong list chro...
Sandip Das, Sk Samim Islam, A. Mohapatra et al.· 0 citations
A proper conflict-free coloring of a graph is a proper vertex coloring in which every non-isolated vertex has a color appearing exactly once in its neighborhood. A graph $G$ is proper conflict-free $(\mathrm{degree}+2)$-choosable if every list assignment $L$ with $|L(v)|\ge d_G(v)+2$ for each $v\in V(G)$ admits such a...
Let $G$ be a graph with minimum degree $\delta(G)\ge|V(G)|/2$. Can we color the edges of $G$ with red and blue so that every pair of non-adjacent vertices is connected by a path consisting of exactly one red edge and one blue edge? We provide an affirmative answer to this question for a class of graphs that are ``close...
J'anos Bar'at, Simona Boyadzhiyska, Andrea Freschi· 0 citations
Let $G$ be a graph. Under a strong edge coloring of $G$, every color class is an induced matching. The strong chromatic index of $G$, denoted by $\chi'_s(G)$, is the smallest integer $k$ such that $G$ admits a strong edge coloring with $k$ colors. Denote by $\Delta(G)$ the maximum degree of $G$. In this paper, we prove...
A vertex-coloring of a graph is centered if every connected subgraph has a vertex with a unique color. A vertex-coloring of a graph is linear if every path in the graph has a vertex with a unique color. Let $\chi_{\mathrm{cen}}(G)$ and $\chi_{\mathrm{lin}}(G)$ be the minimum number of colors in a centered (resp. linear...
Jędrzej Hodor, P. Micek· 0 citations
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