Two conjectures on uniquely totally colorable graphs

  • Authors:

  • S. Akbari

  • Affiliations:

  • Department of Mathematical Sciences, Sharif University of Technology, Tehran 11365, Iran and Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran

  • Venue:
  • Discrete Mathematics - Special issue: The 18th British combinatorial conference
  • Year:
  • 2003

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Abstract

In this paper we investigate two conjectures proposed in (Graphs Combin. 13 (1997) 305-314). The first one is uniquely totally colorable (UTC) conjecture which states: Empty graphs, paths, and cycles of order 3k, k a natural number, are the only UTC graphs. We show that if G is a UTC graph of order n, then Δ ≤ n/2 + 1, where Δ is the maximum degree of G. Also there is another question about UTC graphs that appeared in (Graphs Combin. 13 (1997) 305-314) as follows: If a graph G is UTC, is it true that in the proper total coloring of G, each color is used for at least one vertex? We prove that if G is a UTC graph of order n and in the proper total coloring of G, there exists a color which did not appear in any vertex of G, then G is a Δ-regular graph and n/2 ≤ Δ ≤ n/2 + 1.