4,806 research outputs found
Number of cliques in graphs with a forbidden subdivision
We prove that for all positive integers , every -vertex graph with no
-subdivision has at most cliques. We also prove that
asymptotically, such graphs contain at most cliques, where
tends to zero as tends to infinity. This strongly answers a question
of D. Wood asking if the number of cliques in -vertex graphs with no
-minor is at most for some constant .Comment: 10 pages; to appear in SIAM J. Discrete Mat
A chain theorem for sequentially -rank-connected graphs with respect to vertex-minors
Tutte (1961) proved the chain theorem for -connected graphs with respect
to minors, which states that every -connected graph has a -connected
minor with one vertex fewer than , unless is a wheel graph. Bouchet
(1987) proved an analog for prime graphs with respect to vertex-minors. We
present a chain theorem for higher connectivity with respect to vertex-minors,
showing that every sequentially -rank-connected graph has a sequentially
-rank-connected vertex-minor with one vertex fewer than , unless
.Comment: 21 page
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