On d-panconnected tournaments with large semidegrees
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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|≤1/2(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≤1/3(p-5) (respectively, m≤1/5(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 1/3(p+1) and 2/3(p-2) (respectively, between 1/5(2p-1) and 1/5(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.
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