Weighted Max Cut in Subcubic Graphs
For a graph $G$ with nonnegative edge weights, let $\operatorname{mc}(G,w)$ denote the maximum total weight of a cut and let $w(G)$ denote the total edge weight. We prove that every edge-weighted triangle-free subcubic graph satisfies $\operatorname{mc}(G,w)\geq \frac{15}{19}w(G)$, improving the bound of $\frac{8}{11}$ due to Gutin and Yeo. For subcubic graphs of odd girth at least seven and nine, we strengthen this bound to $\frac{47}{59}w(G)$ and $\frac45w(G)$, respectively, establishing the conjecture of Gutin and Yeo for graphs of odd girth at least nine. Our proofs combine structural properties of vertex partitions with randomized cut constructions and convex combinations of the resulting bounds.
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