Pancyclism in the Generalized Sum of Digraphs

TL;DR

This paper investigates the pancyclic properties of strong digraphs formed as generalized sums of disjoint Hamiltonian digraphs, establishing conditions under which these sums are pancyclic, vertex-pancyclic, or Hamiltonian with specific cycle lengths.

Contribution

It provides new theoretical results characterizing the cycle structures of generalized sums of Hamiltonian digraphs, extending understanding of pancyclicity in complex digraph constructions.

Findings

Strong generalized sums of two Hamiltonian digraphs are either pancyclic, vertex-pancyclic, or Hamiltonian with specific cycle lengths.

For collections of k disjoint Hamiltonian digraphs, the generalized sum exhibits similar cycle properties with bounds depending on subset sums.

The paper establishes conditions ensuring the existence of cycles of all lengths within certain bounds in these complex digraphs.

Abstract

A digraph $D=(V,A)$ of order $n\geq 3$ is pancyclic, whenever $D$ contains a directed cycle of length $k$ for each $k\in\{3,...,n\}$; and D is vertex-pancyclic iff, for each vertex $v\in V$ and each $k\in \{3,...,n\}$, $D$ contains a directed cycle of length $k$ passing through $v$. Let $D_1$, $D_2$,..., $D_k$ be a collection of pairwise vertex disjoint digraphs. The generalized sum (g.s.) of $D_1$, $D_2$,..., $D_k$, denoted by $\oplus_{i=1}^k D_i$ or $D_1\oplus D_2 \oplus \cdots \oplus D_k$, is the set of all digraphs D satisfying: (i) $V(D)=\bigcup_{i=1}^k V(D_i)$, (ii) $D\langle V(D_i) \rangle \cong D_i$ for $i=1, 2,..., k$; and (iii) for each pair of vertices belonging to different summands of D, there is exactly one arc between them, with an arbitrary but fixed direction. A digraph $D\in \oplus_{i=1}^k D_i$ will be called a generalized sum (g.s.) of $D_1$, $D_2$,..., $D_k$. In this paper we prove that if $D_1$ and $D_2$ are two vertex disjoint Hamiltonian digraphs and $D\in D_1 \oplus D_2$ is strong, then at least one of the following assertions holds: $D$ is vertex-pancyclic, it is pancyclic or it is Hamiltonian and contains a directed cycle of length $l$ for each $l\in\{3,..., \max\{|V(D_i)|+1 \colon i\in\{1,2\}\}\}$. Moreover, we prove that if $D_1$, $D_2$,..., $D_k$ is a collection of pairwise vertex disjoint Hamiltonian digraphs, $n_i=|V(D_i)|$ for each $i\in \{1,...,k\}$ and $D \in \oplus_{i=1}^k D_i$ is strong, then at least one of the following assertions holds: $D$ is vertex-pancyclic, it is pancyclic or it is Hamiltonian and contains a directed cycle of length $l$ for each $l\in \{3,..., \max\{\left(\sum_{i\in S} n_i\right) + 1 \colon S\subset\{1,..., k\}\text{ with }|S|=k-1\}\}$.