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Lower Bounds for Maximum Weight Bisections of Graphs with Bounded Degrees

Stefanie Gerke, Gregory Gutin, Anders Yeo, Yacong Zhou

TL;DR

The paper addresses lower bounds for maximum weight bisections in edge-weighted graphs with bounded maximum degree, presenting a parity-based conjecture and proving several key cases. It blends probabilistic methods with equitable coloring, Vizing-type decompositions, and structural graph analysis (including bridgeless and triangle-free subcubic graphs) to derive concrete bounds. Notably, it establishes a $\frac{2}{3}$-factor for weighted subcubic graphs and a $\frac{613}{855}$-factor for bridgeless triangle-free subcubic graphs, while outlining extensions to broader classes and posing conjectures (including a $\frac{11}{15}$ bound) for triangle-free cases. These results advance the understanding of bisection weights under degree constraints and guide future work on tight bounds and conjectures for special graph families.

Abstract

A bisection in a graph is a cut in which the number of vertices in the two parts differ by at most 1. In this paper, we give lower bounds for the maximum weight of bisections of edge-weighted graphs with bounded maximum degree. Our results improve a bound of Lee, Loh, and Sudakov (J. Comb. Th. Ser. B 103 (2013)) for (unweighted) maximum bisections in graphs whose maximum degree is either even or equals 3, and for almost all graphs. We show that a tight lower bound for maximum size of bisections in 3-regular graphs obtained by Bollobás and Scott (J. Graph Th. 46 (2004)) can be extended to weighted subcubic graphs. We also consider edge-weighted triangle-free subcubic graphs and show that a much better lower bound (than for edge-weighted subcubic graphs) holds for such graphs especially if we exclude $K_{1,3}$. We pose three conjectures.

Lower Bounds for Maximum Weight Bisections of Graphs with Bounded Degrees

TL;DR

The paper addresses lower bounds for maximum weight bisections in edge-weighted graphs with bounded maximum degree, presenting a parity-based conjecture and proving several key cases. It blends probabilistic methods with equitable coloring, Vizing-type decompositions, and structural graph analysis (including bridgeless and triangle-free subcubic graphs) to derive concrete bounds. Notably, it establishes a -factor for weighted subcubic graphs and a -factor for bridgeless triangle-free subcubic graphs, while outlining extensions to broader classes and posing conjectures (including a bound) for triangle-free cases. These results advance the understanding of bisection weights under degree constraints and guide future work on tight bounds and conjectures for special graph families.

Abstract

A bisection in a graph is a cut in which the number of vertices in the two parts differ by at most 1. In this paper, we give lower bounds for the maximum weight of bisections of edge-weighted graphs with bounded maximum degree. Our results improve a bound of Lee, Loh, and Sudakov (J. Comb. Th. Ser. B 103 (2013)) for (unweighted) maximum bisections in graphs whose maximum degree is either even or equals 3, and for almost all graphs. We show that a tight lower bound for maximum size of bisections in 3-regular graphs obtained by Bollobás and Scott (J. Graph Th. 46 (2004)) can be extended to weighted subcubic graphs. We also consider edge-weighted triangle-free subcubic graphs and show that a much better lower bound (than for edge-weighted subcubic graphs) holds for such graphs especially if we exclude . We pose three conjectures.
Paper Structure (6 sections, 22 theorems, 36 equations, 7 figures)

This paper contains 6 sections, 22 theorems, 36 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Conjecture \ref{['cj:1']} for even $k$ and almost all graphs
  3. Conjecture \ref{['cj:1']} for weighted subcubic graphs
  4. Triangle-free Subcubic Graphs
  5. Bridgress triangle-free subcubic graphs
  6. Extending bridgeless case to all cases except a claw

Key Result

Theorem 1.2

LLS If $G$ has maximum degree at most $k$, then there exists a bisection of size at least $\frac{k+1}{2k}|E(G)|-\frac{k(k+1)}{4}$ if $k$ is odd and $\frac{k+2}{2(k+1)}|E(G)|-\frac{k(k+2)}{4}$ if $k$ is even.

Figures (7)

  • Figure 1: Balanced bipartite graph with an additional vertex
  • Figure 2: $H[N_3\cup N_2\cup V_2(H)]$
  • Figure 3: On the left $C$ with a chord, on the right graphs $B_C$
  • Figure 4: On the left $C$ with two chords, on the right graphs $B_C$
  • Figure 5: On the left $C$ with three chords, on the right graphs $B_C$
  • ...and 2 more figures

Theorems & Definitions (49)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • ...and 39 more