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Two conjectures on uniquely totally colorable graphs

Akbari, S ; Sharif University of Technology | 2003

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  1. Type of Document: Article
  2. DOI: 10.1016/S0012-365X(02)00797-5
  3. Publisher: 2003
  4. Abstract:
  5. 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 Δn/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/2Δn/2+1. © 2003 Elsevier Science B.V. All rights reserved
  6. Keywords:
  7. Graph coloring ; Total coloring ; Uniquely ; UTC
  8. Source: Discrete Mathematics ; Volume 266, Issue 1-3 , 2003 , Pages 41-45 ; 0012365X (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/pii/S0012365X02007975