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On small uniquely vertex-colourable graphs and Xu's conjecture [electronic resource]

Daneshgar, A. (Amir) ; Sharif University of Technology

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  1. Type of Document: Article
  2. DOI: 10.1016/S0012-365X(00)00042-X
  3. Abstract:
  4. Consider the parameter Λ(G) = |E(G)| - |V(G)|(k - 1) + (k2) for a k-chromatic graph G, on the set of vertices V(G) and with the set of edges E(G). It is known that Λ(G)≥0 for any k-chromatic uniquely vertex-colourable graph G (k-UCG), and, S.J. Xu has conjectured that for any k-UCG, G, Λ(G) = 0 implies that cl(G) = k; in which cl(G) is the clique number of G. In this paper, first, we introduce the concept of the core of a k-UCG as an induced subgraph without any colour-class of size one, and without any vertex of degree k - 1. Considering (k, n)-cores as k-UCGs on n vertices, we show that edge-minimal (k, 2k)-cores do not exist when k ≥ 3, which shows that for any edge-minimal k-UCG on 2k vertices either the conjecture is true or there exists a colour-class of size one. Also, we consider the structure of edge-minimal (k, 2k + 1)-cores and we show that such cores exist for all k ≥ 4. Moreover, we characterize all edge-minimal (4,9)-cores and we show that there are only seven such cores (up to isomorphism). Our proof shows that Xu's conjecture is true in the case of edge-minimal (4,9)-cores
  5. Keywords:
  6. Graph colouring ; Uniquely vertex-colourable graphs ; Xu's conjecture
  7. Source: Discrete Mathematics ; Volume 223, Issues 1–3, 28 August 2000, Pages 93–108
  8. URL: http://www.sciencedirect.com/science/article/pii/S0012365X0000042X