Euler dynamic H-trails in edge-colored graphs

Abstract

Alternating Euler trails has been extensively studied for its diverse practical and theoretical applications. Let H be a graph possibly with loops and G be a multigraph without loops. In this paper we deal with any fixed coloration of E(G) with V(H) (H-coloring of G). A sequence W = ( v 0 , e 0 1 , … , e 0 k 0 , v 1 , e 1 1 , … , e n − 1 k n − 1 , v n ) in G, where for each i ∈ { 0 , … , n − 1 } , k i ≥ 1 and e i j = v i v i + 1 is an edge in G, for every j ∈ { 1 , … , k i } , is a dynamic H-trail if W does not repeat edges and c ( e i k i ) c ( e i + 1 1 ) is an edge in H, for each i ∈ { 0 , … , n − 2 } . In particular, a dynamic H-trail is an alternating trail when H is a complete graph without loops and ki = 1, for every i ∈ { 1 , … , n − 1 } . In this paper, we introduce the concept of dynamic H-trail, which arises in a natural way in the modeling of many practical problems, in particular, in theoretical computer science. We provide necessary and sufficient conditions for the existence of closed Euler dynamic H-trail in H-colored multigraphs. Also we provide polynomial time algorithms that allows us to convert a cycle in an auxiliary graph, L 2 H ( G ) , in a closed dynamic H-trail in G, and vice versa, where L 2 H ( G ) is a non-colored simple graph obtained from G in a polynomial time.