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Isolation of regular graphs and $k$-chromatic graphs

Peter Borg

TL;DR

The paper studies the F-isolation number for families defined by regularity and chromatic number, introducing $\mathcal{F}_{1,k}$, $\mathcal{F}_{2,k}$, and $\mathcal{F}_{3,k}$ and proving a sharp universal bound $\iota(G, \mathcal{F}_{i,k}) \le \frac{m+1}{\binom{k}{2}+2}$ for connected $m$-edge graphs that are not $k$-cliques. Equality is achieved by a broad class of extremal graphs described by the pure $(m,k)$-special construction, with a full characterization of when equality holds, including small exceptional cases. The authors extend these results to the $k$-clique and cycle isolation settings and provide a unified framework that connects to prior sharp bounds in the literature. The work introduces a new feature in isolation by determining and leveraging the extremal graphs that realize the bound, and it yields practical, exact bounds for related isolation problems.

Abstract

Given a set $\mathcal{F}$ of graphs, we call a copy of a graph in $\mathcal{F}$ an $\mathcal{F}$-graph. The $\mathcal{F}$-isolation number of a graph $G$, denoted by $ι(G,\mathcal{F})$, is the size of a smallest set $D$ of vertices of $G$ such that the closed neighbourhood of $D$ intersects the vertex sets of the $\mathcal{F}$-graphs contained by $G$ (equivalently, $G - N[D]$ contains no $\mathcal{F}$-graph). Thus, $ι(G,\{K_1\})$ is the domination number of $G$. For any integer $k \geq 1$, let $\mathcal{F}_{1,k}$ be the set of regular graphs of degree at least $k-1$, let $\mathcal{F}_{2,k}$ be the set of graphs whose chromatic number is at least $k$, and let $\mathcal{F}_{3,k}$ be the union of $\mathcal{F}_{1,k}$ and $\mathcal{F}_{2,k}$. Thus, $k$-cliques are members of both $\mathcal{F}_{1,k}$ and $\mathcal{F}_{2,k}$. We prove that for each $i \in \{1, 2, 3\}$, $\frac{m+1}{{k \choose 2} + 2}$ is a best possible upper bound on $ι(G, \mathcal{F}_{i,k})$ for connected $m$-edge graphs $G$ that are not $k$-cliques. The bound is attained by infinitely many (non-isomorphic) graphs. The proof of the bound depends on determining the graphs attaining the bound. This appears to be a new feature in the literature on isolation. Among the result's consequences are a sharp bound of Fenech, Kaemawichanurat and the present author on the $k$-clique isolation number and a sharp bound on the cycle isolation number.

Isolation of regular graphs and $k$-chromatic graphs

TL;DR

The paper studies the F-isolation number for families defined by regularity and chromatic number, introducing , , and and proving a sharp universal bound for connected -edge graphs that are not -cliques. Equality is achieved by a broad class of extremal graphs described by the pure -special construction, with a full characterization of when equality holds, including small exceptional cases. The authors extend these results to the -clique and cycle isolation settings and provide a unified framework that connects to prior sharp bounds in the literature. The work introduces a new feature in isolation by determining and leveraging the extremal graphs that realize the bound, and it yields practical, exact bounds for related isolation problems.

Abstract

Given a set of graphs, we call a copy of a graph in an -graph. The -isolation number of a graph , denoted by , is the size of a smallest set of vertices of such that the closed neighbourhood of intersects the vertex sets of the -graphs contained by (equivalently, contains no -graph). Thus, is the domination number of . For any integer , let be the set of regular graphs of degree at least , let be the set of graphs whose chromatic number is at least , and let be the union of and . Thus, -cliques are members of both and . We prove that for each , is a best possible upper bound on for connected -edge graphs that are not -cliques. The bound is attained by infinitely many (non-isomorphic) graphs. The proof of the bound depends on determining the graphs attaining the bound. This appears to be a new feature in the literature on isolation. Among the result's consequences are a sharp bound of Fenech, Kaemawichanurat and the present author on the -clique isolation number and a sharp bound on the cycle isolation number.
Paper Structure (2 sections, 7 theorems, 34 equations)

This paper contains 2 sections, 7 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['mainresult2']}

Key Result

Theorem 1

If $k \geq 1$ and $G$ is a connected $m$-edge graph that is not a $k$-clique, then Moreover: (i) Equality in (cliquebnd) holds if and only if either $G$ is a pure $(m,k)$-special graph or $k=2$ and $G$ is a $5$-cycle. (ii) If $G$ is an $(m,k)$-special graph, then $\iota(G,K_k) = \left \lfloor (m+1)/\left( {k\choose2}+2 \right ) \right \rfloor.$

Theorems & Definitions (7)

  • Theorem 1: BFK2
  • Theorem 2
  • Theorem 3
  • Lemma 1: Borg
  • Lemma 2: Borg
  • Proposition 1
  • Theorem 4: Brooks' Theorem Brooks