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Edge Clique Cover of Claw-free Graphs
The smallest number of cliques, covering all edges of a graph , is
called the (edge) clique cover number of and is denoted by . It
is an easy observation that for every line graph with vertices,
. G. Chen et al. [Discrete Math. 219 (2000), no. 1--3, 17--26;
MR1761707] extended this observation to all quasi-line graphs and questioned if
the same assertion holds for all claw-free graphs. In this paper, using the
celebrated structure theorem of claw-free graphs due to Chudnovsky and Seymour,
we give an affirmative answer to this question for all claw-free graphs with
independence number at least three. In particular, we prove that if is a
connected claw-free graph on vertices with , then and equality holds if and only if is either the graph of
icosahedron, or the complement of a graph on vertices called twister or
the power of the cycle , for .Comment: 74 pages, 4 figure
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