Abstract
A conjecture by Erds and Hajnal [Ramsey-type theorems. Discrete Appl Math. 1989;25:37–52] pointed out that the structure of graphs with forbidden induced subgraphs is very different from general graphs. More precisely, they contain much larger cliques or independent sets. Inspired by this, we prove that non-Hamilton-connected graphs satisfying certain conditions contain a large clique. From the perspective of looking for a maximum clique, we present some sufficient conditions for a graph to be Hamilton-connected in terms of size, spectral radius and signless Laplacian spectral radius, respectively, which extend some corresponding results.
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