Search

In 1962, Erdős gave a sufficient condition for Hamilton cycles in terms of the vertex number, edge number and minimum degree of graphs which generalized Ore’s theorem. One year later, Moon and Moser gave an analogous result for Hamilton cycles in balanced bipartite graphs. In this paper, we present the spectral analogues of Erdős’ theorem and Moon–Moser’s theorem, respectively. Letbe the class of non-Hamiltonian graphs of ordernand minimum degree at leastk. We determine the maximum (signless Laplacian) spectral radius of graphs in(for large enoughn), and the minimum (signless Laplacian) spectral radius of the complements of graphs in. All extremal graphs with the maximum (signless Laplacian) spectral radius and with the minimum (signless Laplacian) spectral radius of the complements are determined, respectively. We also solve similar problems for balanced bipartite graphs and the quasi-complements. [ABSTRACT FROM PUBLISHER]

Let t 1 , … , t r ∈ [ 4 , 2 q ] be any r even integers, where q ≥ 2 and r ≥ 1 are two integers. In this note, we show that every bipartite tournament with minimum outdegree at least q r − 1 contains r vertex-disjoint directed cycles of lengths t 1 ′ , … , t r ′ such that t i ′ = t i for t i = 0 ( mod 4 ) and t i ′ ∈ { t i , t i + 2 } for t i = 2 ( mod 4 ) , where 1 ≤ i ≤ r . The special case q = 2 of the result verifies the bipartite tournament case of a conjecture proposed by Bermond and Thomassen, stating that every digraph with minimum outdegree at least 2 r − 1 contains at least r vertex-disjoint directed cycles. [ABSTRACT FROM AUTHOR]