On properly ordered coloring of vertices in a vertex-weighted graph

TL;DR

This paper introduces properly ordered coloring (POC) for weighted graphs, exploring its properties, bounds, and extensions of classical theorems, thereby generalizing vertex coloring concepts to weighted graphs.

Contribution

It defines POC for weighted graphs, establishes key functions and bounds, and extends the Gallai-Hasse-Roy-Vitaver theorem to weighted contexts.

Findings

f(G) equals the length of the longest path in G

Bound on ratio of χ_POC(G;t) - 1 to χ(G) - 1 by t

Determined χ_POC(G;t) for complete multipartite graphs

Abstract

We introduce the notion of a properly ordered coloring (POC) of a weighted graph, that generalizes the notion of vertex coloring of a graph. Under a POC, if $xy$ is an edge, then the larger weighted vertex receives a larger color; in the case of equal weights of $x$ and $y$, their colors must be different. In this paper, we shall initiate the study of this special coloring in graphs. For a graph $G$, we introduce the function $f(G)$ which gives the maximum number of colors required by a POC over all weightings of $G$. We show that $f(G)=\ell(G)$, where $\ell(G)$ is the number of vertices of a longest path in $G$. Another function we introduce is $\chi_{POC}(G;t)$ giving the minimum number of colors required over all weightings of $G$ using $t$ distinct weights. We show that the ratio of $\chi_{POC}(G;t)-1$ to $\chi(G)-1$ can be bounded by $t$ for any graph $G$; in fact, the result is shown by determining $\chi_{POC}(G;t)$ when $G$ is a complete multipartite graph. We also determine the minimum number of colors to give a POC on a vertex-weighted graph in terms of the number of vertices of a longest directed path in an orientation of the underlying graph. This extends the so called Gallai-Hasse-Roy-Vitaver theorem, a classical result concerning the relationship between the chromatic number of a graph $G$ and the number of vertices of a longest directed path in an orientation of $G$.