Long Path Lemma concerning Connectivity and Independence Number

Long Path Lemma concerning Connectivity and Independence Number

We show that, in a k-connected graph G of order n with α(G) = α, between any pair of vertices, there exists a path P joining them with |P | ≥ min { n, (k−1)(n−k) α + k } . This implies that, for any edge e ∈ E(G), there is a cycle containing e of length at least min { n, (k−1)(n−k) α + k } . Moreover, we generalize our result as follows: for any choice S of s ≤ k vertices in G, there exists a t...

Long path connectivity of regular graphs

Gauss's Lemma for Number Fields

Complete Minors and Independence Number

Let G be a graph with n vertices and independence number α. Hadwiger’s conjecture implies that G contains a clique minor of order at least n/α. In 1982, Duchet and Meyniel proved that this bound holds within a factor 2. Our main result gives the first improvement on their bound by an absolute constant factor. We show that G contains a clique minor of order larger than .504n/α. We also prove rel...

On an Inequality of Sagher and Zhou concerning Stein’s Lemma

The jth Rademacher function rj on [0, 1), j = 0, 1, 2, . . . , is defined as follows: r0 = 1, r1 = 1 on [0, 1/2) and r1 = −1 on [1/2, 1), r2 = 1 on [0, 1/4) ∪ [1/2, 3/4) and r2 = −1 on [1/4, 1/2) ∪ [3/4, 1), etc. The following is a classical result that can be found in Zygmund [10] (page 213): For every subset E of [0, 1] and every λ > 1, there is a positive integer N such that for all complex-...