Transforming eulerian trails

Fleischner, H., G. Sabidussi and E. Wenger, Transforming eulerian trails, Discrete Mathematics 109 (1992) 103-116. In this paper a set of transformations (K-transformations) between eulerian trails is investigated. It is known that two arbitrary eulerian trails can be transformed into each other by a sequence of K-transformations. For compatible eulerran trails the set of K-transformations is augmented by the set of K-detachments and K-absorptions. This augmented set is capable of transforming two arbitrary P-compatible eulerian trails (P is an edge partition system) into each other. This result is applied to A-trails, alternating eulerian trails and digraphs. Introduction and preliminaries Given any two eulerian trails T, and T2 of a connected eulerian graph G, it is known that T2 can be obtained from T, by a sequence of K-transformations (see Definition 2 below); this has been shown by various authors (see, e.g., [l, 71). In the present paper we consider eulerian trails of G which satisfy certain restrictions regarding their transitions. Our aim is to develop a type of transformations which, on the one hand, yields a result analogous to the above, and, on the other, respects the restrictions on the corresponding sets of eulerian trails. For this purpose, K-transformations are not sufficiently general (this has been noted already in [S]), and other types of transformations have to be introduced. As a consequence of this approach, we also obtain a method to transform any two eulerian trails of a weakly connected eulerian diagraph into each other. For notation and terminology not defined in this paper see [2,3]. Let G be a graph, by V(G) we denote the set of all vertices of G, by F(G) c V(G) the set of all vertices v of G with d(v) = i, i E N, where d(v) is the degree of r~. By E(G) we denote the set of all edges of G and by E,,(G) c E(G) the set of all edges of G incident with TV E V(G). For the definition of transition systems f or graphs with multiple edges we need the concept of half-edges. Each edge e incident with TV and u.’ consists of two 0012-365X/92/$05.00 @ 1992Elsevier Science Publishers B.V. All rights reserved 104 H. Fleischner ci al. half-edges e( “I and et”‘), e(“) being incident with v and the other incident with w. By EL(G) we denote the set of all half-edges incident with v E V(6). For the sake of simplicity of notation we will not distinguish between edges and half-edges. In particular, we will use the same symbols for edges and half-edges and omit the tedious superscripts of half-edges. For a graph G ard 1~ E V2, we call a partition P(v) of E:(G) a partition systers? 2: u. P(G):= u PO-0 UEV(G).d(U)>2 is called a partition system of G. If each set in P(G) has exactly two elements P(G) is called a transition system of G. For a connected eulerian graph G, T(G) denotes the set of all eulerian trails of G, and .sP,I(G) denotes the set of all decompositions of G into exactly n closed trails. If T is a closed trail in a graph G, we define XT, the transition system of T, by XT := {{ei, cj} 13~ E V(G) with ei, ej E E:(G) and ei, V. ej is a subsequence of T}. Similarly we define the transition system X5 of a decomposition S = {S,, * . , Sn) of G into closed trails S’ by Xs := Ui &,. For TV E V(G), XT(v j c XT (Xs(v) c Xs) denotes the set of transitions at the vertex ‘u (i.e., transitions with half-edges incident with v).