On group choosability of total graphs

TL;DR

This paper explores group and list group colorings of total graphs, proposing two conjectures and confirming the group version of the total coloring conjecture for various graph classes.

Contribution

It introduces the group versions of total and list total coloring conjectures and proves these for multiple classes of graphs, expanding understanding of graph colorings.

Findings

Established the group total coloring conjecture for graphs with small maximum degree

Proved the group list total coloring conjecture for forests and outerplanar graphs

Confirmed the group total coloring conjecture for two-degenerate and certain planar graphs

Abstract

In this paper, we study the group and list group colorings of total graphs and we give two group versions of the total and list total colorings conjectures. We establish the group version of the total coloring conjecture for the following classes of graphs: graphs with small maximum degree, two-degenerate graphs, planner graphs with maximum degree at least 11, planner graphs without certain small cycles, outerplanar and near-outerplanar graphs. In addition, the group version of the list total coloring conjecture is established for forests, outerplanar graphs and graphs with maximum degree at most two.