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The existence of biregular spanning subgraphs in bipartite graphs via spectral radius

Dandan Fan, Xiaofeng Gu, Huiqiu Lin

Abstract

Biregular bipartite graphs have been proven to have similar edge distributions to random bipartite graphs and thus have nice pseudorandomness and expansion properties. Thus it is quite desirable to find a biregular bipartite spanning subgraph in a given bipartite graph. In fact, a theorem of Ore implies a structural characterization of such subgraphs in bipartite graphs. In this paper, we demonstrate the existence of biregular bipartite spanning subgraphs in bipartite graphs by employing spectral radius. We also study the existence of spanning trees with restricted degrees and edge-disjoint spanning trees in bipartite graphs via spectral radius.

The existence of biregular spanning subgraphs in bipartite graphs via spectral radius

Abstract

Biregular bipartite graphs have been proven to have similar edge distributions to random bipartite graphs and thus have nice pseudorandomness and expansion properties. Thus it is quite desirable to find a biregular bipartite spanning subgraph in a given bipartite graph. In fact, a theorem of Ore implies a structural characterization of such subgraphs in bipartite graphs. In this paper, we demonstrate the existence of biregular bipartite spanning subgraphs in bipartite graphs by employing spectral radius. We also study the existence of spanning trees with restricted degrees and edge-disjoint spanning trees in bipartite graphs via spectral radius.

Paper Structure

This paper contains 5 sections, 15 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The proofs of Theorems \ref{['thm::biregular-factor']}, \ref{['thm::spanning-tree1']} and \ref{['thm::spanning-tree2']}
  4. The proof of Theorem \ref{['thm::edge-spanningtree']}
  5. Concluding remarks

Key Result

Theorem 1.1

Suppose that $a$ and $b$ are two positive integers with $b>a$. Let $G$ be a connected $(X,Y)$-bipartite graph with $am=bn$, where $|X|=m$ and $|Y|=n$. If $m\geq n+b$ and then $G$ contains an $(a,b)$-biregular factor, unless $G\cong K_{m,n}\backslash E(K_{1,n-a+1})$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1: See Nosal
  • Lemma 2.2: Brouwer and Haemers BH, p. 30; Godsil and Royle C.Godsil, pp. 196--198
  • Lemma 3.1
  • proof
  • Theorem 3.2: Ore Ore57
  • ...and 13 more