On 2-bisections and monochromatic edges in claw-free cubic multigraphs

Abstract

A $k$-bisection of a multigraph $G$ is a partition of its vertex set into two parts of the same cardinality such that every component of each part has at most $k$ vertices. Cui and Liu shown that every claw-free cubic multigraph contains a $2$-bisection, while Eom and Ozeki constructed specific $2$-bisections with bounded number of monochromatic edges. Their bound is the best possible for claw-free cubic simple graphs. In this note, we extend the latter result to the larger family of claw-free cubic multigraphs

Figures (3)

• Figure 1: From left to right: a digon, a triangle, a trumpet and a diamond.
• Figure 2: The two possible cases when the new edge $xy$ was already an edge of $G$.
• Figure 3: The $2$-bisections $(B^{\prime},W^{\prime})$ of $G^{\prime}$ and $(B,W)$ of $G$, when the number of diamonds $k$ is odd.

Theorems & Definitions (8)

• Conjecture 1.1: banlinial
• Theorem 1.2: cuiliu
• Lemma 2.1
• Definition 2.2
• Lemma 2.3: kenta
• Theorem 2.4: kenta
• Theorem 3.1
• proof