Degree Conditions for Dominating Cycles in 1-tough Graphs

Details Degree Conditions for Dominating Cycles in 1-tough Graphs Degree Conditions for Dominating Cycles in 1-tough Graphs

2012-01-30

arxiv.org/abs/1201.6330v2

Mathematics

Combinatorics

27 pages

Scientific paper

We prove: (i) if $G$ is a 1-tough graph of order $n$ and minimum degree $\delta$ with $\delta\ge(n-2)/3$ then each longest cycle in $G$ is a dominating cycle unless $G$ belongs to an easily specified class of graphs with $\kappa(G)=2$ and $\tau(G)=1$. The second result follows immediately from the first result: (ii) if $G$ is a 3-connected 1-tough graph with $\delta\ge(n-2)/3$ then each longest cycle in $G$ is a dominating cycle.

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