Edge-partitioning graphs into regular and locally irregular components

International audienceA graph is locally irregular if every two adjacent vertices have distinct degrees. Recently, Baudon et al. introduced the notion of decomposition into locally irregular subgraphs. They conjectured that for almost every graph $G$, there exists a minimum integer $\chi^{\prime}_{\mathrm{irr}}(G)$ such that $G$ admits an edge-partition into $\chi^{\prime}_{\mathrm{irr}}(G)$ classes, each of which induces a locally irregular graph. In particular, they conjectured that $\chi^{\prime}_{\mathrm{irr}}(G) \leq 3$ for every $G$, unless $G$ belongs to a well-characterized family of non-decomposable graphs. This conjecture is far from being settled, as notably (1) no constant upper bound on$\chi^{\prime}_{\mathrm{irr}}(G)$ is known for $G$ bipartite, and (2) no satisfactory general upper bound on $\chi^{\prime}_{\mathrm{irr}}(G)$ is known. We herein investigate the consequences on this question of allowing a decomposition to include regular components as well. As a main result, we prove that every bipartite graph admits such a decomposition into at most $6$ subgraphs. This result implies that every graph $G$ admits a decomposition into at most $6(\lfloor \mathrm{log} \chi (G) \rfloor +1)$ subgraphs whose components are regular or locally irregular

Decomposability of graphs into subgraphs fulfilling the 1-2-3 Conjecture

The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every $d$-regular graph, $d\geq 2$, can be decomposed into at most $2$ subgraphs (without isolated edges) fulfilling the 1-2-3 Conjecture if $d\notin\{10,11,12,13,15,17\}$, and into at most $3$ such subgraphs in the remaining cases. Additionally, we prove that in general every graph without isolated edges can be decomposed into at most $24$ subgraphs fulfilling the 1-2-3 Conjecture, improving the previously best upper bound of $40$. Both results are partly based on applications of the Lov\'asz Local Lemma.Comment: 13 page

Locally irregular edge-coloring of subcubic graphs

A graph is {\em locally irregular} if no two adjacent vertices have the same degree. A {\em locally irregular edge-coloring} of a graph $G$ is such an (improper) edge-coloring that the edges of any fixed color induce a locally irregular graph. Among the graphs admitting a locally irregular edge-coloring, i.e., {\em decomposable graphs}, only one is known to require $4$ colors, while for all the others it is believed that $3$ colors suffice. In this paper, we prove that decomposable claw-free graphs with maximum degree $3$, all cycle permutation graphs, and all generalized Petersen graphs admit a locally irregular edge-coloring with at most $3$ colors. We also discuss when $2$ colors suffice for a locally irregular edge-coloring of cubic graphs and present an infinite family of cubic graphs of girth $4$ which require $3$ colors

Decomposing degenerate graphs into locally irregular subgraphs

International audienceA (undirected) graph is locally irregular if no two of its adjacent vertices have the same degree. A decomposition of a graph G into k locally irregular subgraphs is a partition E_1,...,E_k of E(G) into k parts each of which induces a locally irregular subgraph. Not all graphs decompose into locally irregular subgraphs; however, it was conjectured that, whenever a graph does, it should admit such a decomposition into at most three locally irregular subgraphs. This conjecture was verified for a few graph classes in recent years.This work is dedicated to the decomposability of degenerate graphs with low degeneracy. Our main result is that decomposable k-degenerate graphs decompose into at most 3k+1 locally irregular subgraphs, which improves on previous results whenever k≤9. We improve this result further for some specific classes of degenerate graphs, such as bipartite cacti, k-trees, and planar graphs. Although our results provide only little progress towards the leading conjecture above, the main contribution of this work is rather the decomposition schemes and methods we introduce to prove these results