Regular colorings and factors of regular graphs

TL;DR

This paper characterizes 4-regular pseudographs lacking (3,1)-colorings, explores conditions for 5-regular pseudographs without (4,1)-colorings or {4,1}-factors, and constructs non-colorable graphs for all r ≥ 6.

Contribution

It provides a complete characterization of 4-regular pseudographs without (3,1)-colorings and introduces new constructions of non-colorable graphs for higher degrees.

Findings

Characterization of 4-regular pseudographs without (3,1)-colorings

Conditions for 5-regular pseudographs lacking (4,1)-colorings or {4,1}-factors

Construction of non-(r-1,1)-colorable graphs for all r ≥ 6

Abstract

An $(r-1,1)$-coloring of an $r$-regular graph $G$ is an edge coloring such that each vertex is incident to $r-1$ edges of one color and $1$ edge of a different color. In this paper, we completely characterize all $4$-regular pseudographs (graphs that may contain parallel edges and loops) which do not have a $(3,1)$-coloring. An $\{r-1,1\}$-factor of an $r$-regular graph is a spanning subgraph in which each vertex has degree either $r-1$ or $1$. We prove various conditions that that must hold for any vertex-minimal $5$-regular pseudographs without $(4,1)$-colorings or without $\{4,1\}$-factors. Finally, for each $r\geq 6$ we construct graphs that are not $(r-1,1)$-colorable and, more generally, are not $(r-t,t)$-colorable for small $t$.