On d-panconnected tournaments with large semidegrees

Abstract

We prove the following new results.(a)Let T be a regular tournament of order 2n+1≥11 and S a subset of V(T). Suppose that |S|≤12(n−2) and x, y are distinct vertices in V(T)∖S. If the subtournament T−S contains an (x,y)-path of length r, where 3≤r≤|V(T)∖S|−2, then T−S also contains an (x,y)-path of length r+1.(b)Let T be an m-irregular tournament of order p, i.e., |d+(x)−d−(x)|≤m for every vertex x of T. If m≤13(p−5) (respectively, m≤15(p−3)), then for every pair of vertices x and y, T has an (x,y)-path of any length k, 4≤k≤p−1 (respectively, 3≤k≤p−1 or T belongs to a family G of tournaments, which is defined in the paper). In other words, (b) means that if the semidegrees of every vertex of a tournament T of order p are between 13(p+1) and 23(p−2) (respectively, between 15(2p−1) and 15(3p−4)), then the claims in (b) hold.Our results improve in a sense related results of Alspach (1967), Jacobsen (1972), Alspach et al. (1974), Thomassen (1978) and Darbinyan (1977, 1978, 1979), and are sharp in a sense.