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

Labeling of graphs, sumset of squares of units modulo n and resonance varieties of matroids

This thesis investigates problems in a number of different areas of graph theory and its applications in other areas of mathematics. Motivated by the 1-2-3-Conjecture, we consider the closed distinguishing number of a graph G, denoted by dis[G]. We provide new upper bounds for dis[G] by using the Combinatorial Nullstellensatz. We prove that it is NP-complete to decide for a given planar subcubic graph G, whether dis[G] = 2. We show that for each integer t there is a bipartite graph G such that dis[G] \u3e t. Then some polynomial time algorithms and NP-hardness results for the problem of partitioning the edges of a graph into regular and/or locally irregular subgraphs are presented. We then move on to consider Johnson graphs to find resonance varieties of some classes of sparse paving matroids. The last application we consider is in number theory, where we find the number of solutions of the equation x21 + _ _ _ + x2 k = c, where c 2 Zn, and xi are all units in the ring Zn. Our approach is combinatorial using spectral graph theory

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