Totally Silver Graphs

TL;DR

This paper introduces the concept of totally silver graphs, providing characterizations, infinite families, and specific examples including cubic graphs with girth up to 10, linking to broader graph theory topics.

Contribution

It offers new constructive characterizations and infinite families of totally silver graphs, including bipartite and cubic cases with specified girth.

Findings

Characterizations of totally silver graphs

Infinite families of such graphs

Existence of cubic totally silver graphs with girth up to 10

Abstract

A totally silver coloring of a graph G is a k--coloring of G such that for every vertex v \in V(G), each color appears exactly once on N[v], the closed neighborhood of v. A totally silver graph is a graph which admits a totally silver coloring. Totally silver coloring are directly related to other areas of graph theory such as distance coloring and domination. In this work, we present several constructive characterizations of totally silver graphs and bipartite totally silver graphs. We give several infinite families of totally silver graphs. We also give cubic totally silver graphs of girth up to 10.