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Spectral extremal problems for $(a,b,k)$-critical and fractional $(a,b,k)$-critical graphs

Zengzhao Xu, Ligong Wang, Weige Xi

TL;DR

The paper addresses spectral radius conditions that guarantee the existence of $(a,b,k)$-critical and fractional $(a,b,k)$-critical graphs, generalizing $[a,b]$-factors. It introduces the extremal construction $F_{n}^{a,b,k}$ and proves that, for large $n$ and $ abla(G)\\ge a+k$, a graph with $lambda(G)\u0003\ge\lambda(F_{n}^{a,b,k})$ is either $(a,b,k)$-critical or isomorphic to the extremal graph, with a parallel size condition. It extends these results to fractional versions, including the fractional $(r,k)$-critical case, yielding analogous spectral thresholds and extremal characterizations; setting $k=0$ recovers open problems on $[a,b]$-factors and fractional $[a,b]$-factors. Collectively, the work advances spectral extremal graph theory for generalized factor problems and provides optimality via explicit extremal constructions, paving avenues for further research on related conjectures and fractional variants.

Abstract

A factor of a graph is essentially a specific type spanning subgraph. The study of characterizing the existence of $[a, b]$-factors based on eigenvalue conditions can be traced back to the work of Brouwer and Haemers (2005) on perfect matchings. With the advancement of graphs factor theory, the related spectral extremal problems, particularly the study of $[a,b]$-factors and fractional $[a,b]$-factors, have been widely studied by scholars. Our work is motivated by research related to the $[a,b]$-factors and fractional $[a,b]$-factors, and explores their generalizations: $(a,b,k)$-critical graphs and fractional $(a,b,k)$-critical graphs. A graph $G$ is called an $(a,b,k)$-critical (a fractional $(a,b,k)$-critical) graph if after deleting any $k$ vertices of $G$ the remaining graph of $G$ has an $[a,b]$-factor (a fractional $[a,b]$-factor). In this paper, we establish spectral radius conditions for a graph to be $(a,b,k)$-critical or fractional $(a,b,k)$-critical. When $k=0$, our results also resolve some open problems concerning $[a, b]$-factors and fractional $[a, b]$-factors.

Spectral extremal problems for $(a,b,k)$-critical and fractional $(a,b,k)$-critical graphs

TL;DR

The paper addresses spectral radius conditions that guarantee the existence of -critical and fractional -critical graphs, generalizing -factors. It introduces the extremal construction and proves that, for large and , a graph with is either -critical or isomorphic to the extremal graph, with a parallel size condition. It extends these results to fractional versions, including the fractional -critical case, yielding analogous spectral thresholds and extremal characterizations; setting recovers open problems on -factors and fractional -factors. Collectively, the work advances spectral extremal graph theory for generalized factor problems and provides optimality via explicit extremal constructions, paving avenues for further research on related conjectures and fractional variants.

Abstract

A factor of a graph is essentially a specific type spanning subgraph. The study of characterizing the existence of -factors based on eigenvalue conditions can be traced back to the work of Brouwer and Haemers (2005) on perfect matchings. With the advancement of graphs factor theory, the related spectral extremal problems, particularly the study of -factors and fractional -factors, have been widely studied by scholars. Our work is motivated by research related to the -factors and fractional -factors, and explores their generalizations: -critical graphs and fractional -critical graphs. A graph is called an -critical (a fractional -critical) graph if after deleting any vertices of the remaining graph of has an -factor (a fractional -factor). In this paper, we establish spectral radius conditions for a graph to be -critical or fractional -critical. When , our results also resolve some open problems concerning -factors and fractional -factors.
Paper Structure (8 sections, 21 theorems, 97 equations, 2 figures)

This paper contains 8 sections, 21 theorems, 97 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Terminology and notation
  3. Background and results
  4. Structure and Organization
  5. Preliminaries
  6. Proofs of Theorems \ref{['T1']} and \ref{['T2']}
  7. Proofs of Theorems \ref{['T3']} and \ref{['T4']}
  8. Concluding Remarks and Future Work

Key Result

Theorem 1.6

For integers $b> a\ge 1$ and $k\ge 0$, and let $G$ be a connected graph of order $n \geq 2(b+a+k + 2)(b+k + 2)$ with minimum degree $\delta(G)\ge a+k$. If then $G$ is an $(a,b,k)$-critical graph, unless $G \cong F_{n}^{a,b,k}$.($F_{n}^{a,b,k}$ is shown in Figure 1).

Figures (2)

  • Figure 1: Graphs $F_{n}^{a,b,k}$
  • Figure 2: Graphs $\mathscr{F}_{n}^{a,b,k}$

Theorems & Definitions (28)

  • Conjecture 1.1
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 18 more