An Extension of the Chvátal–Erdős Theorem: Counting the Number of Maximum Independent Sets

Abstract

Chvátal and Erdős proved a well-known result that the graph $G$ with connectivity $\kappa (G)$ not less than its independence number $\alpha (G)$ [$\alpha (G)+1$, $\alpha (G)-1$, respectively] is Hamiltonian (traceable, Hamiltonian-connected, respectively). In this paper, we strengthen the Chvátal–Erdős theorem to the following: Let $G$ be a simple 2-connected graph of order large enough such that $\alpha (G)\le \kappa (G)+1$ [$\alpha (G)\le \kappa (G)+2$, $\alpha (G)\le \kappa (G),$ respectively] and such that the number of maximum independent sets of cardinality $\kappa (G)+1$ [$\kappa (G)+2$, $\kappa (G)$, respectively] is at most $n-2\kappa (G)$ [$n-2\kappa (G)-1$, $n-2\kappa (G)+1$, respectively]. Then $G$ is either Hamiltonian (traceable, Hamiltonian-connected, respectively) or a subgraph of $K_{k}+((kK_1)\cup K_{n-2k})$ [$K_{k}+((k+1)K_1\cup K_{n-2k-1})$, $K_{k}+((k-1)K_1\cup K_{n-2k+1})$, respectively].