Spanning Trees with a Bounded Number of Branch Vertices in a Claw-Free Graph

نویسندگان

Haruhide Matsuda

Kenta Ozeki

Tomoki Yamashita

چکیده

Let k be a non-negative integer. A branch vertex of a tree is a vertex of degree at least three. We show two sufficient conditions for a connected clawfree graph to have a spanning tree with a bounded number of branch vertices: (i) A connected claw-free graph has a spanning tree with at most k branch vertices if its independence number is at most 2k + 2. (ii) A connected clawfree graph of order n has a spanning tree with at most one branch vertex if the degree sum of any five independent vertices is at least n− 2. These conditions are best possible. A related conjecture also is proposed.

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