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On total chromatic number of direct product graphs

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Abstract

Coloring of the graph products, especially vertex and edge coloring, has been widely researched for all types of graph products. For total graph coloring, as combination of edge and vertex coloring, Behzad and Vizing set Total Coloring Conjecture in mid 1960s. In this paper, we prove the conjecture for two specific direct graph products, for direct product of path and arbitrary graph G, P n ×G, where χ′(G)=Δ(G), and expand the proof onto direct product of arbitrary cycle and a path P n , C m ×P n . At the same time, the proofs provide the algorithms to color such graphs.

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Authors and Affiliations

  1. Institute for Mathematics, Physics and Mechanics, Jadranska 19, 1000, Ljubljana, Slovenia

    Katja Prnaver

  2. Faculty of Mathematics and Natural Sciences, University of Maribor, Koroška c. 160, 2000, Maribor, Slovenia

    Blaž Zmazek

Authors
  1. Katja Prnaver
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  2. Blaž Zmazek
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Correspondence to Katja Prnaver.

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Prnaver, K., Zmazek, B. On total chromatic number of direct product graphs. J. Appl. Math. Comput. 33, 449–457 (2010). https://doi.org/10.1007/s12190-009-0296-8

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  • DOI: https://doi.org/10.1007/s12190-009-0296-8

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