Vertex Alternating-Pancyclism in 2-Edge-Colored Graphs

TL;DR

This paper investigates conditions under which 2-edge-colored graphs, constructed as generalized sums of disjoint subgraphs, contain alternating cycles of all lengths passing through each vertex, extending the concept of pancyclism.

Contribution

It provides sufficient conditions for a graph in the colored generalized sum to be vertex alternating-pancyclic, advancing understanding of cycle structures in 2-edge-colored graphs.

Findings

Identifies conditions ensuring the existence of all-length alternating cycles through each vertex.

Extends the theory of pancyclism to 2-edge-colored graphs with generalized sum structures.

Offers applications in areas requiring diverse cycle structures in colored graphs.

Abstract

An alternating cycle in a 2-two-edge-colored graph is a cycle such that any two consecutive edges have different colors. Let $G_1, \ldots, G_k$ be a collection of pairwise vertex disjoint 2-edge-colored graphs. The colored generalized sum of $G_1, \ldots, G_k$, denoted by $ \oplus_{i=1}^k G_i$, is the set of all 2-edge-colored graphs $G$ such that: (i) $V(G)=\bigcup_{i=1}^k V(G_i)$, (ii) $G\langle V(G_i)\rangle\cong G_i$ for $i=1,\ldots, k$ as edge-colored graphs where $G\langle V(G_i)\rangle$ has the same coloring as $G_i$ and (iii) between each pair of vertices in different summands of $G$ there is exactly one edge, with an arbitrary but fixed color. A graph $G$ in $\oplus_{i=1}^k G_i$ will be called a colored generalized sum (c.g.s.) and we will say that $e\in E(G)$ is an exterior edge iff $e\in E(G)\setminus \left(\bigcup_{i=1}^k E(G_i)\right)$. The set of exterior edges will be denoted by $E_\oplus$. A colored graph $G$ is said to be a vertex alternating-pancyclic graph, whenever for each vertex $v$ in $G$, and for each $l\in\{3,\ldots, |V(G)|\}$, there exists in $G$ an alternating cycle of length $l$ passing through $v$. The topics of pancyclism and vertex-pancyclism are deeply and widely studied by several authors. The existence of alternating cycles in 2-edge-colored graphs has been studied because of its many applications. In this paper, we give sufficient conditions for a graph $G\in \oplus_{i=1}^k G_i$ to be a vertex alternating-pancyclic graph.