Decomposing graphs into a constant number of locally irregular subgraphs

Julien Bensmail 1 Martin Merker 2 Carsten Thomassen 2
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : A graph is locally irregular if no two adjacent vertices have the same degree. The irregular chromatic index chi_irr'(G) of a graph G is the smallest number of locally irregular subgraphs needed to edge-decompose G. Not all graphs have such a decomposition, but Baudon, Bensmail, Przybyło, and Woźniak conjectured that if G can be decomposed into locally irregular subgraphs, then chi_irr'(G)⩽3. In support of this conjecture, Przybyło showed that chi_irr'(G)⩽3 holds whenever G has minimum degree at least 10^10. Here we prove that every bipartite graph G which is not an odd length path satisfies chi_irr'(G)⩽10. This is the first general constant upper bound on the irregular chromatic index of bipartite graphs. Combining this result with Przybyło's result, we show that chi_irr'(G)⩽328 for every graph G which admits a decomposition into locally irregular subgraphs. Finally, we show that chi_irr'(G)⩽2 for every 16-edge-connected bipartite graph G.
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Julien Bensmail, Martin Merker, Carsten Thomassen. Decomposing graphs into a constant number of locally irregular subgraphs. European Journal of Combinatorics, Elsevier, 2017, 60, pp.124 - 134. ⟨10.1016/j.ejc.2016.09.011⟩. ⟨hal-01629938⟩

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