Relating the independence number and the dissociation number

TL;DR

This paper investigates relationships between the independence number and dissociation number in graphs, providing improved inequalities for specific graph classes and characterizing extremal graphs.

Contribution

It establishes new bounds relating independence and dissociation numbers for various graph classes, including cubic and bipartite graphs, and characterizes extremal cases.

Findings

Connected cubic graphs (except K4) satisfy 5α(G) ≥ 3 diss(G).

Improved bounds for bipartite and triangle-free graphs.

Characterization of extremal subcubic graphs where inequalities are tight.

Abstract

The independence number $\alpha(G)$ and the dissociation number ${\rm diss}(G)$ of a graph $G$ are the largest orders of induced subgraphs of $G$ of maximum degree at most $0$ and at most $1$, respectively. We consider possible improvements of the obvious inequality $2\alpha(G)\geq {\rm diss}(G)$. For connected cubic graphs $G$ distinct from $K_4$, we show $5\alpha(G)\geq 3{\rm diss}(G)$, and describe the rich and interesting structure of the extremal graphs in detail. For bipartite graphs, and, more generally, triangle-free graphs, we also obtain improvements. For subcubic graphs though, the inequality cannot be improved in general, and we characterize all extremal subcubic graphs.