Graphs with coloring redundant edges

Abstract

A graph edge is $d$-coloring redundant if the removal of the edge doesnot change the set of $d$-colorings of the graph. Graphs that are toosparse or too dense do not have coloring redundant edges. Tight upperand lower bounds on the number of edges in a graph in order for thegraph to have a coloring redundant edge are proven. Two constructionslink the class of graphs with a coloring redundant edge to the$K_4$-free graphs and to the uniquely colorable graphs. The structureof graphs with a coloring redundant edge is explored.

Keywords

• coloring of graphs and hypergraphs, extremal problems