Edge colorings and circular flows on regular graphs

Davide Mattiolo,Eckhard Steffen

doi:10.1002/jgt.22746

Abstract

AbstractLet be the circular flow number of a bridgeless graph . By Steffen it was proved that, for a bridgeless ‐regular graph , where is a positive integer, if and only if has a perfect matching such that is bipartite. This implies that is a class 1 graph. For , all graphs with circular flow number bigger than 4 are class 2 graphs. We show that, for all , . This was conjectured to be true by Steffen. Moreover we prove that, for all , is a ‐regular class 1 graph with no perfect matching whose removal leaves a bipartite graph. We further disprove the conjecture that every ‐regular class 1 graph has circular flow number at most .

Highlights

A coloring c is proper if c(e1) = c(e2) for any two adjacent edges e1, e2 ∈ E(G)
The chromatic index χ′(G) of G is the minimum k such that G admits a proper k-edgecoloring and it is known that χ′(G) ∈ {∆(G), . . . , ∆(G) + μ(G)}, where μ(G) denotes the maximum multiplicity of an edge in G [14]
Since (2t + 1)-regular class 1 graphs are (2t + 1)-graphs Conjecture 1.5 implies Conjecture 1.4. We show that both these conjectures are false by constructing (2t+1)-regular class 1 graphs with circular flow number greater than 2 + 2t

Summary

Introduction

In this paper we consider finite graphs G with vertex set V (G) and edge set E(G). A graph may contain multiple edges but no loops. Let us define G2t+1 := {G : G is a (2t+1)-regular class 1 graph such that there is no perfect matching M of G such that G − M is bipartite}. Let H be a cubic graph with a perfect matching M3 such that, for all positive integers t, H + (2t − 2)M3 is a (2t + 1)-regular class 1 For i = 1, 2, let Gi be a cubic graph having the Mi-class-2 property, where Mi is a perfect matching of Gi. let G be an (M1, M2)dot-product of G1 and G2 and x, y ∈ V (G2) the two adjacent vertices that have been removed from G2 when constructing G.

Let n

Perform the dot product

Construction of

Construction of Mp