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Forcing structures and cliques in uniquely vertex colorable graphs

Daneshgar, A ; Sharif University of Technology | 2001

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  1. Type of Document: Article
  2. DOI: 10.1137/S0895480196304994
  3. Publisher: 2001
  4. Abstract:
  5. Let G be a simple undirected uniquely vertex k-colorable graph, or a k-UCG for short. M. Truszczyński [Some results on uniquely colorable graphs, in Finite and Infinite Sets, North-Holland, Amsterdam, 1984, pp. 733-748] introduced e* (G) = |V(G)|(k - 1) - (k2) as the minimum number of edges for a k-UCG and S. J. Xu [J. Combin. Theory Ser. B, 50 (1990), pp. 319-320] conjectured that any minimal A-UCG contains a Kk as a subgraph. In this paper, first we introduce a technique called forcing. Then by applying this technique in conjunction with a feedback structure we construct a k-UCG with clique number k - t, for each t ≥ 1 and each k, when k is large enough. This also improves some known results for the case t = 1. Second, we analyze the parameter Λ(G) = |E(G)| - e* (G) for our constructions, and we obtain some bounds for the functions λt(k) = min{Λ(G) : G is a k-UCG and cl(G) = k - t}, νt(k) = min{|V(G)| : G is a k-UCG and cl(G) = k - t}. Also, we introduce some possible applications of the technique in cryptography and data compression
  6. Keywords:
  7. Uniquely vertex colorable graphs
  8. Source: SIAM Journal on Discrete Mathematics ; Volume 14, Issue 4 , 2001 , Pages 433-445 ; 08954801 (ISSN)
  9. URL: https://epubs.siam.org/doi/10.1137/S0895480196304994