Elsevier

Discrete Mathematics

Volume 310, Issues 15–16, 28 August 2010, Pages 2082-2090
Discrete Mathematics

The Chvátal–Erdös condition for supereulerian graphs and the Hamiltonian index

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Abstract

A classical result of Chvátal and Erdös says that the graph G with connectivity κ(G) not less than its independence number α(G) (i.e. κ(G)α(G)) is Hamiltonian. In this paper, we show that the 2-connected graph G with κ(G)α(G)1 is one of the following: supereulerian, the Petersen graph, the 2-connected graph with three vertices of degree two obtained from K2,3 by replacing a vertex of degree three with a triangle, one of the 2-connected graphs obtained from K2,3 by replacing a vertex of degree two with a complete graph of order at least three and by replacing at most one branch of length two in the resulting graph with a branch of length three, or one of the graphs obtained from K2,3 by replacing at most two branches of K2,3 with a branch of length three. We also show that the Hamiltonian index of the simple 2-connected graph G with κ(G)α(G)t is at most 2t+23 for every nonnegative integer t. The upper bound is sharp.

Keywords

Connectivity
Independence number
Supereulerian graph
Hamiltonian index

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