Euler dynamic H-trails in edge-colored graphs

TL;DR

This paper introduces the concept of dynamic H-trails in edge-colored graphs, providing necessary and sufficient conditions for their existence and polynomial algorithms for their construction, extending the theory of alternating Euler trails.

Contribution

It defines dynamic H-trails, generalizing alternating Euler trails, and offers new theoretical conditions and polynomial algorithms for their detection and construction.

Findings

Necessary and sufficient conditions for closed Euler dynamic H-trails.

Polynomial algorithms to convert cycles in auxiliary graphs to dynamic H-trails.

Extension of alternating Euler trail theory to more general edge-colored graphs.

Abstract

Alternating Euler trails has been extensively studied for its diverse applications, for example, in genetic and molecular biology, social science and channel assignment in wireless networks, as well as for theoretical reasons. We will consider the following edge-coloring. Let $H$ be a graph possibly with loops and $G$ a graph without loops. An $H$-coloring of $G$ is a function $c: E(G) \rightarrow V(H)$. We will say that $G$ is an $H$-colored graph whenever we are taking a fixed $H$-coloring of $G$. A sequence $W=(v_0,e_0^1, \ldots, e_0^{k_0},v_1,e_1^1,\ldots,e_{n-1}^{k_{n-1}},v_n)$ in $G$, where for each $i \in \{0,\ldots, n-1\}$, $k_i \geq 1$ and $e_i^j = v_iv_{i+1}$ is an edge in $G$, for every $j \in \{1,\ldots, k_i \}$, 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 $i \in \{0,\ldots,n-2\}$. In particular a dynamic $H$-trail is an alternating Euler trail when $H$ is a complete graph without loops and $k_i=1$, for every $i \in \{1,\ldots,n-1\}$. 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.