Antidirected Hamilton circuits and paths in tournaments

Abstract

A tournament is a directed graph such that each pair of vertices VI, V2 is joined by exactly one of the edges (VI, V2), (V2, V 0. The set of vertices and the set of edges of the directed graph G are denoted ~:(G) and ¢(G) respectively. If (1:1, I:2) e 8(G) we shall say that V 1 dominates V 2. A directed graph is transitive whenever (V1, 1:2) e ¢(G) and (1:2, V3) e ¢(G) implies (V1, 1:3)e 8(G). If T is a tournament and ~¢ _~ ~U(7) then T(~¢) denotes the subtournament of T spanned by ~¢. Trev denotes the tournament obtained from T by reversing the direction of every edge of T. If V~, V2, ..., Vk is a finite sequence of mutually distinct vertices of the directed graph G and (~, V~÷I) e 8(G) for 1 < i < k 1 then the vertices V1, V 2 . . . . . V k together with the edges (V l, [ /2) , ( V 2 , I/3) , . . . , ( V k _ l , Vk) a r e called a directed path. If (V~, V~+I)e g(G) for i odd and (Vi+l, V~)e 8(G) for i even then the vertices Vt, V2,..., Vk together with the edges (I:1, V2), (V3, V2), (Va, 1:4) . . . . are called an antidirected path. This path will be denoted V~ --. 1/2 ~ I:3 ~ "-'. V1 is called a starting vertex and Vk is called a starting/(ending) vertex of this path if k is odd/(even). If furthermore Vt dominates Vk and k is even then the vertices I:1, V2,..., Vk together with the edges (Vt, V2), (Va, V2) . . . . . (Vk-~, Vk), (1:1, Vk) are called an antidirected circuit. Similarly we define an antidirected path with two ending vertices, a one-way infinite antidirected path with starting (or ending) vertex and a two-way infinite antidirected path. A path/(circuit) which includes all vertices of the directed graph G is called a Hamilton path/ (circuit) of G. An ADH path/(circuit) is an antidirected Hamilton path/(circuit). Every finite tournament has a directed Hamilton path (see e.g. [2], p. 30) and there exist enumerably infinite tournaments which have no directed Hamilton path (e.g. the tournament which consists of the vertices Vo, V1, 1:2 . . . . and the edges (V~, Vo) for i = 1, 2, 3 . . . . and all the edges (V~, Vj) with 1 < i <j). Every finite tournament with three exceptions has an ADH path ([1]). We shall prove the following theorems: