Symmetric Set Coloring of Signed Graphs

TL;DR

This paper introduces a unifying symmetric set coloring concept for signed graphs, establishing theoretical bounds, defining a new chromatic number, and connecting it to existing coloring frameworks.

Contribution

It proposes a new symmetric set coloring framework for signed graphs, proves bounds, and introduces the symset-chromatic number linking colorings to graph partitions.

Findings

Proves a Brooks'-type theorem for symmetric set colorings.

Defines the symset-chromatic number and relates it to graph partitions.

Shows the connection between symmetric set colorings and DP-colorings.

Abstract

There are many concepts of signed graph coloring which are defined by assigning colors to the vertices of the graphs. These concepts usually differ in the number of self-inverse colors used. We introduce a unifying concept for this kind of coloring by assigning elements from symmetric sets to the vertices of the signed graphs. In the first part of the paper, we study colorings with elements from symmetric sets where the number of self-inverse elements is fixed. We prove a Brooks'-type theorem and upper bounds for the corresponding chromatic numbers in terms of the chromatic number of the underlying graph. These results are used in the second part where we introduce the symset-chromatic number $\chi_{sym}(G,\sigma)$ of a signed graph $(G,\sigma)$. We show that the symset-chromatic number gives the minimum partition of a signed graph into independent sets and non-bipartite antibalanced subgraphs. In particular, $\chi_{sym}(G,\sigma) \leq \chi(G)$. In the final section we show that these colorings can also be formalized as $DP$-colorings.