Forcing Structures and Cliques in Uniquely Vertex Colorable Graphs

Abstract

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 $e^{^{*}}(G)=|V(G)|(k-1)-{k \choose 2}$ 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 k-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 $t \geq 1$ and each k, when k is large enough. This also improves some known results for the case t=1. Second, we analyze the parameter $\Lambda(G)=|E(G)|-e^{^{*}}(G)$ for our constructions, and we obtain some bounds for the functions \lambda_{_{t}}(k)= \min \{\Lambda(G) \ : \ G \ {\rm is \ a} \ k\mbox{\rm -UCG and} \ cl(G)=k-t \}, $$ $$ \nu_{_{t}}(k)= \min \{|V(G)| \ : \ G \ {\rm is \ a} \ k\mbox{\rm -UCG and} \ cl(G)=k-t \}. Also, we introduce some possible applications of the technique in cryptography and data compression.