[Paper Review] Cycles and p-competition graphs

Suh-Ryung KIMBoram Park, Park, Boram+1 more|arXiv (Cornell University)|May 25, 2009

Advanced Graph Theory Research6 references3 citations

TL;DR

This paper characterizes when a cycle $C_n$ and its complement $¯C_n$ are $p$-competition graphs by determining exact values of $p$ in terms of $n$. It proves that $C_n$ is a $p$-competition graph if and only if $n \geq p+3$, and provides sufficient conditions for $¯C_n$ to be a $p$-competition graph based on $n$ and $p$, generalizing prior results on competition graphs and edge clique covers.

ABSTRACT

The notion of p-competition graphs of digraphs was introduced by S-R. Kim, T. A. McKee, F. R. McMorris, and F. S. Roberts [p-competition graphs, Linear Algebra Appl., 217 (1995) 167--178] as a generalization of the competition graphs of digraphs. Let p be a positive integer. The p-competition graph C_p(D) of a digraph D=(V,A) is a (simple undirected) graph which has the same vertex set V and has an edge between distinct vertices x and y if and only if there exist p distinct vertices v_1, ..., v_p in V such that (x,v_i), (y,v_i) are arcs of the digraph D for each i=1, ..., p. In this paper, given a cycle of length n, we compute exact values of p in terms of n such that it is a p-competition graph, which generalizes the results obtained by Kim et al. We also find values of p in terms of n so that its complement is a p-competition graph.

Motivation & Objective

• To extend the theory of $p$-competition graphs by characterizing when cycles $C_n$ are $p$-competition graphs.
• To determine sufficient conditions under which the complement of a cycle $\overline{C_n}$ is a $p$-competition graph.
• To generalize prior results on $2$-competition graphs of cycles by Kim et al. to arbitrary $p$.
• To analyze the $p$-edge clique cover number $\theta_e^p(G)$ as a key tool for characterizing $p$-competition graphs.

Proposed method

• Uses Theorem 1.1 to characterize $p$-competition graphs via the existence of a $p$-edge clique cover of size at most $n$.
• Applies contradiction arguments to show that $C_n$ cannot be a $p$-competition graph if $p \geq n-2$, establishing the lower bound $n \geq p+3$.
• Constructs explicit $p$-edge clique covers for $\overline{C_n}$ using structured clique families based on parity and modular arithmetic of vertex indices.
• Analyzes edge coverage in $\overline{C_n}$ by classifying edges based on parity of endpoints and distance, assigning them to specific cliques in the cover.
• Uses Theorem 1.2 to derive sufficient conditions for $\overline{C_n}$ to be a $p$-competition graph from the edge clique cover number $\theta_e(\overline{C_n})$.
• Employs case analysis for odd and even $n$ to construct minimal edge clique covers and bound $\theta_e(\overline{C_n})$.

Experimental results

Research questions

• RQ1 For which values of $p$ and $n$ is the cycle $C_n$ a $p$-competition graph?
• RQ2 What is the minimum $p$ such that $C_n$ fails to be a $p$-competition graph?
• RQ3 For which $p$ and $n$ is the complement $\overline{C_n}$ a $p$-competition graph?
• RQ4 How does the edge clique cover number $\theta_e(\overline{C_n})$ relate to the $p$-competition graph property of $\overline{C_n}$?
• RQ5 Can the $p$-edge clique cover number $\theta_e^p(G)$ be used to fully characterize $p$-competition graphs of cycles?

Key findings

• A cycle $C_n$ is a $p$-competition graph if and only if $n \geq p + 3$.
• For $n \geq 9$ odd, $\overline{C_n}$ is a $p$-competition graph if $p \leq \frac{n-3}{2}$.
• For $n \geq 10$ even, $\overline{C_n}$ is a $p$-competition graph if $p \leq \frac{n}{2}$.
• The complement $\overline{C_8}$ has edge clique cover number $\theta_e(\overline{C_8}) = 6$, and is a $p$-competition graph for $p \leq 5$.
• The complement $\overline{C_6}$ is a $p$-competition graph for $p \leq 4$, and $\overline{C_7}$ for $p \leq 1$.
• Explicit constructions of $p$-edge clique covers for $\overline{C_n}$ are provided for both odd and even $n$, proving the sufficiency of the derived bounds.