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We determine the maximum number of edges that a claw-free graph can have, when its maximum degree and matching number are bounded. This is a famous problem that has been studied on general graphs, and for which there is a tight bound. The graphs achieving this bound contain in most cases an induced copy of K 1 , 3 , the claw, which motivates studying the question on claw-free graphs. Note that...
A connected graph G with chromatic number t is double-critical if G ∖ { x , y } is ( t − 2 ) -colorable for each edge x y ∈ E ( G ) . The complete graphs are the only known examples of double-critical graphs. A long-standing conjecture of Erdős and Lovász from 1966, which is referred to as the Double-Critical Graph Conjecture, states that there are no other double-critical graphs. That...
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