Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A precise condition for independent transversals in bipartite covers

Stijn Cambie, Penny Haxell, Ross J. Kang, Ronen Wdowinski

TL;DR

The paper provides a sharp, bipartite-refined criterion for the existence of independent transversals in bipartite-cover graphs: if the cover degrees $D_A$ and $D_B$ and the list sizes $k_A$ and $k_B$ satisfy $\frac{D_B}{k_A}+\frac{D_A}{k_B}\le 1$, then an independent transversal exists. This result consolidates and extends prior IT results (notably by Haxl and Szabó–Tardos) and is shown to be best possible by constructing full $(k_A,k_B,D_A,D_B)$-graphs with no IT when the ratio exceeds 1. The argument comprises a general corollary to known IT theorems plus a three-phase constructive proof of sharpness, culminating in a comprehensive asymmetric counterexample framework. The work also connects to broader colour-degree questions and situates the results within the landscape of correspondence-covers and list-covers, highlighting both the strengths and limits of the bipartite refinement. Overall, the paper clarifies how part sizes and maximum degrees interact to guarantee ITs and delineates the sharp boundary of this phenomenon.

Abstract

Given a bipartite graph $H=(V=V_A\cup V_B,E)$ in which any vertex in $V_A$ (resp.~$V_B$) has degree at most $D_A$ (resp.~$D_B$), suppose there is a partition of $V$ that is a refinement of the bipartition $V_A\cup V_B$ such that the parts in $V_A$ (resp.~$V_B$) have size at least $k_A$ (resp.~$k_B$). We prove that the condition $D_A/k_B+D_B/k_A\le 1$ is sufficient for the existence of an independent set of vertices of $H$ that is simultaneously transversal to the partition, and show moreover that this condition is sharp. This result is a bipartite refinement of two well-known results on independent transversals, one due to the second author and the other due to Szabó and Tardos.

A precise condition for independent transversals in bipartite covers

TL;DR

The paper provides a sharp, bipartite-refined criterion for the existence of independent transversals in bipartite-cover graphs: if the cover degrees and and the list sizes and satisfy , then an independent transversal exists. This result consolidates and extends prior IT results (notably by Haxl and Szabó–Tardos) and is shown to be best possible by constructing full -graphs with no IT when the ratio exceeds 1. The argument comprises a general corollary to known IT theorems plus a three-phase constructive proof of sharpness, culminating in a comprehensive asymmetric counterexample framework. The work also connects to broader colour-degree questions and situates the results within the landscape of correspondence-covers and list-covers, highlighting both the strengths and limits of the bipartite refinement. Overall, the paper clarifies how part sizes and maximum degrees interact to guarantee ITs and delineates the sharp boundary of this phenomenon.

Abstract

Given a bipartite graph in which any vertex in (resp.~) has degree at most (resp.~), suppose there is a partition of that is a refinement of the bipartition such that the parts in (resp.~) have size at least (resp.~). We prove that the condition is sufficient for the existence of an independent set of vertices of that is simultaneously transversal to the partition, and show moreover that this condition is sharp. This result is a bipartite refinement of two well-known results on independent transversals, one due to the second author and the other due to Szabó and Tardos.
Paper Structure (8 sections, 11 theorems, 7 equations, 2 figures)

This paper contains 8 sections, 11 theorems, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A sufficient condition
  3. Sharpness of the condition
  4. Phase I.
  5. Phase II.
  6. Phase III.
  7. Sharpness in a very asymmetric setting
  8. Open access statement.

Key Result

Theorem 1.3

Let $H=H(A,B,G,L)$ be a bipartite-cover. Let positive integers $k_A$, $k_B$, $D_A$, $D_B$ be such that $\frac{D_B}{k_A} + \frac{D_A}{k_B} \le 1$. If the maximum degrees in $A_H$ and $B_H$ are $D_A$ and $D_B$, respectively, and $|L(v)| \ge k_A$ for all $v \in A_G$ and $|L(w)| \ge k_B$ for all $w \in

Figures (2)

  • Figure 1: A bipartite-cover graph with maximum degree $3$ and partition classes of size $5$ with no IT. The partition classes are represented by ovals, except that the vertices contained in orange are all a single partition class, as are the vertices contained in blue.
  • Figure 2: A full $(5,6,3,3)$-graph, where the partition classes are represented by ovals, except that the vertices contained in orange are all a single partition class, as are the vertices contained in blue. Note that the graph in Figure 1 is a subgraph of this example.

Theorems & Definitions (25)

  • Conjecture 1.1: Alon and Krivelevich AlKr98
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6: CaKa22
  • Proposition 1.7: CaKa22
  • Conjecture 1.8
  • Theorem 1.9
  • proof
  • Theorem 2.1
  • ...and 15 more