A coloring of the edges of a graph G is strong if each color class is an induced matching of G. The strong chromatic index of G, denoted by χ ′ s (G), is the least number of colors in a strong edge coloring of G. In this note we prove that χ ′ s (G) ≤ (4k − 1)∆(G) − k(2k + 1) + 1 for every k-degenerate graph G. This confirms the strong version of conjecture stated recently by Chang and Narayanan [3]. Our approach allows also to improve the upper bound from [3] for chordless graphs. We get that χ ′ s (G) ≤ 4∆ − 3 for any chordless graph G. Both bounds remain valid for the list version of the strong edge coloring of these graphs.

A vertex coloring φ of a graph G is p-centered if for every connected subgraph H of G either φ uses more than p colors on H or there is a color that appears exactly once on H. Centered colorings form one of the families of parameters that allow to capture notions of sparsity of graphs: A class of graphs has bounded expansion if and only if there is a function f such that for every p 1, every graph in the class admits a p-centered coloring using at most f (p) colors.In this paper, we give upper bounds for the maximum number of colors needed in a p-centered coloring of graphs from several widely studied graph classes. We show that:(1) planar graphs admit p-centered colorings with O(p 3 log p) colors where the previous bound was O(p 19 ); (2) bounded degree graphs admit p-centered colorings with O(p) colors while it was conjectured that they may require exponential number of colors in p;(3) graphs avoiding a fixed graph as a topological minor admit p-centered colorings with a polynomial in p number of colors. All these upper bounds imply polynomial algorithms for computing the colorings. Prior to this work there were no non-trivial lower bounds known. We show that: (4) there are graphs of treewidth t that require p+t t colors in any p-centered coloring and this bound matches the upper bound; (5) there are planar graphs that require Ω(p 2 log p) colors in any p-centered coloring. We also give asymptotically tight bounds for outerplanar graphs and planar graphs of treewidth 3. We prove our results with various proof techniques. The upper bound for planar graphs involves an application of a recent structure theorem while the upper bound for bounded degree graphs comes from the entropy compression method. We lift the result for bounded degree graphs to graphs avoiding a fixed topological minor using the Grohe-Marx structure theorem.

A vertex coloring $\phi$ of a graph $G$ is $p$-centered if for every connected subgraph $H$ of $G$ either $\phi$ uses more than $p$ colors on $H$ or there is a color that appears exactly once on $H$. Centered colorings form one of the families of parameters that allow to capture notions of sparsity of graphs: A class of graphs has bounded expansion if and only if there is a function $f$ such that for every $p\geq1$, every graph in the class admits a $p$-centered coloring using at most $f(p)$ colors. In this paper, we give upper bounds for the maximum number of colors needed in a $p$-centered coloring of graphs from several widely studied graph classes. We show that: (1) planar graphs admit $p$-centered colorings with $O(p^3\log p)$ colors where the previous bound was $O(p^{19})$; (2) bounded degree graphs admit $p$-centered colorings with $O(p)$ colors while it was conjectured that they may require exponential number of colors. All these upper bounds imply polynomial algorithms for computing the colorings. Prior to this work there were no non-trivial lower bounds known. We show that: (4) there are graphs of treewidth $t$ that require $\binom{p+t}{t}$ colors in any $p$-centered coloring. This bound matches the upper bound; (5) there are planar graphs that require $\Omega(p^2\log p)$ colors in any $p$-centered coloring. We also give asymptotically tight bounds for outerplanar graphs and planar graphs of treewidth $3$. We prove our results with various proof techniques. The upper bound for planar graphs involves an application of a recent structure theorem while the upper bound for bounded degree graphs comes from the entropy compression method. We lift the result for bounded degree graphs to graphs avoiding a fixed topological minor using the Grohe-Marx structure theorem.

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 CF . In the literature, this invariant is also called the closed neighborhood conflict-free