Lower Bounds for Maximum Weighted Cut

TL;DR

This work provides a comprehensive treatment of lower bounds for the Maximum Weighted Cut ${mac(G)}$ in weighted graphs. It develops a generic bound via ${\cal B}(G)$-subgraphs, derives DFS-tree and girth-based bounds, and leverages probabilistic methods and Vizing's index to obtain new results for arbitrary graphs and specifically triangle-free graphs with bounded degree. Notably, it proves ${mac(G) \ge \frac{8}{11} w(G)}$ for triangle-free graphs with $\Delta(G) \le 3$ and a fractional tree-bound ${mac(G) \ge \frac{w(G)}{2} + 0.3193 w(T)}$ for the same class with a spanning tree $T$, while also presenting tighter constants and conjectures for subcubic and bounded-degree cases. These contributions advance understanding of weighted cut structures, provide algorithmic lower bounds, and reveal connections to cycle structure and spanning-tree weights in sparse graphs.

Abstract

While there have been many results on lower bounds for Max Cut in unweighted graphs, there are only few results for lower bounds for Max Cut in weighted graphs. In this paper, we launch an extensive study of lower bounds for Max Cut in weighted graphs. We introduce a new approach for obtaining lower bounds for Weighted Max Cut. Using it, Probabilistic Method, Vizing's chromatic index theorem, and other tools, we obtain several lower bounds for arbitrary weighted graphs, weighted graphs of bounded girth and triangle-free weighted graphs. We pose conjectures and open questions.

Figures (5)

• Figure 1: The graph $G'$ in Theorem \ref{['NPdfs']}. The thick edges have weight 2 and the thin edges have weight 1.
• Figure 2: We illustrate the proof of Theorem \ref{['girth']} with the example shown in (a), where we see a DFS tree (thick edges) of a graph $G$. The remaining graphs depict $G_0$, $G_1$, $G_2$ and $G_3$, respectively, where $k=4$.
• Figure 3: Petersen graph with $E'$ in boldface
• Figure 4: Petersen graph $G$ and matching $M$ in boldface
• Figure 5: An illustration of the sets $C_i$'s in Claim F.3.

Theorems & Definitions (49)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Proposition 2.3
• Corollary 2.4
• proof
• Theorem 3.1
• proof
• Corollary 3.2