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arXiv · 1702.02060

Maximizing the number of edges in optimal $k$-rankings

Rigoberto Florez·Darren A. Narayan
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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 $G=P_{2^{k-1}}$, $C_{2^{k}}$, $K_{m_{1},m_{2},\dots,m_{t}}$, 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.

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Rigoberto Florez, Darren A. Narayan. 2017-02-07. Maximizing the number of edges in optimal $k$-rankings. https://arxiv.org/abs/1702.02060

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