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) = ℓ(G), where ℓ(G) is the number of vertices of a longest path in G. Another function we introduce is χPOC(G; t) giving the minimum number of colors required over all weightings of G using t distinct weights. We show that the ratio of χPOC(G; t) − 1 to χ(G) − 1 can be bounded by t for any graph G; in fact, the result is shown by determining χPOC(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.
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Acknowledgments
The authors would like to thank the referees for carefully reading our article and for many helpful comments. The first author’s research was supported by JSPS KAKENHI (19K03603).
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Fujita, S., Kitaev, S., Sato, S. et al. On Properly Ordered Coloring of Vertices in a Vertex-Weighted Graph. Order 38, 515–525 (2021). https://doi.org/10.1007/s11083-021-09554-7
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DOI: https://doi.org/10.1007/s11083-021-09554-7