Colouring Edges with many Colours in Cycles

TL;DR

This paper introduces generalized p-arboricity and p-acyclic edge chromatic numbers, relating them to graph density and cycle properties, extending classical arboricity concepts to multigraphs and proper colourings.

Contribution

It defines and studies generalized p-arboricity and p-acyclic edge chromatic numbers, connecting them to multigraph density and shallow subdivisions, advancing graph colouring theory.

Findings

Established bounds relating p-arboricity to graph density.

Connected p-acyclic edge chromatic numbers with shallow subdivisions.

Extended classical arboricity concepts to multigraphs and proper colourings.

Abstract

The arboricity of a graph G is the minimum number of colours needed to colour the edges of G so that every cycle gets at least two colours. Given a positive integer p, we define the generalized p-arboricity Arb_p(G) of a graph G as the minimum number of colours needed to colour the edges of a multigraph G in such a way that every cycle C gets at least min(|C|; p + 1) colours. In the particular case where G has girth at least p + 1, Arb_p(G) is the minimum size of a partition of the edge set of G such that the union of any p parts induce a forest. If we require further that the edge colouring be proper, i.e., adjacent edges receive distinct colours, then the minimum number of colours needed is the generalized p-acyclic edge chromatic number of G. In this paper, we relate the generalized p-acyclic edge chromatic numbers and the generalized p-arboricities of a graph G to the density of the multigraphs having a shallow subdivision as a subgraph of G.