Maximizing the number of edges in optimal k-rankings

Abstract

A k-ranking is a vertex k-coloring such that if two vertices have the same color any path connecting them contains a vertex of larger color. The rank number of a graph is smallest k such that G has a k-ranking. For certain graphs G we consider the maximum number of edges that may be added to G without changing the rank number. Here we investigate the problem for G = P 2 k−1 , C 2 k , K m 1 ,m 2 ,...,mt , and the union of two copies of K n joined by a single edge. In addition to determining the maximum number of edges that may be added to G without changing the rank number we provide an explicit characterization of which edges change the rank number when added to G, and which edges do not. † Corresponding address: School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY

Extrema Property of the $k$-Ranking of Directed Paths and Cycles

“…Note that when the direction is removed from the edges in this construction, the resulting graph is the graph found in [4] for the undirected case. Also note that this construction of ← − G n is not unique.…”

“…This raises the question "what is the maximum size of a directed graph that satisfies the property that its rank number is equal to the rank number of its largest directed subpath?" Flórez and Narayan [4,5] found results related to this question; however, the problem is still open. We believe that studying particular cases will lead to a better understanding of the problem and potential solutions.…”

“…In this paper we study the necessary and sufficient conditions for the largest possible directed graph that can be obtained by attaching edges to either a directed path or directed cycle without changing the rank number and the number of vertices. In [4] there is a solution for the undirected case. Here, we analyze cases for which the new directed graph keeps the rank number of the original graph.…”