Regular colorings and factors of regular graphs
Anton Bernshteyn, Omid Khormali, Ryan R. Martin, Jonathan Rollin,, Danny Rorabaugh, Songling Shan, Andrew J. Uzzell

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