Independence, induced subgraphs, and domination in $K_{1,r}$-free graphs

TL;DR

This work develops sharp, broadly applicable bounds linking induced-subgraph size to the domination number in $K_{1,r}$-free graphs. It introduces a unifying framework: for a family $\mathcal{F}$ with $\chi(\mathcal{F})\le k$, $\alpha_{\mathcal{F}}(G) \le (r-1)k\gamma(G)$ for all $G\in\mathcal{G}_{r}$, with multiple tight refinements and Ramsey-based extensions $\alpha_{\mathcal{F}}(G) \le r(K_r, \mathcal{F}^*)\gamma(G)$; specialization to $K_q$-free subgraphs yields $\alpha_{\mathcal{F}}(G) \le (r(K_q, K_r)-1)\gamma(G)$. The paper also establishes sharp bounds for independence and $k$-independence numbers in claw-free and regular graphs, providing explicit constructions that achieve equality and infinite families that preserve these ratios. Overall, the results offer a cohesive, Ramsey-theoretic approach to extremal induced-subgraph problems in restricted graph classes, with concrete consequences for independence and domination theory.

Abstract

Let $G$ be a graph and $\mathcal{F}$ a family of graphs. Define $α_{\mathcal{F}}(G)$ as the maximum order of any induced subgraph of $G$ that belongs to the family $\mathcal{F}$. For the family $\mathcal{F}$ of graphs with \emph{chromatic number} at most~$k$, we prove that if $G$ is $K_{1,r}$-free, then $α_{\mathcal{F}}(G) \le (r-1)kγ(G)$, where $γ(G)$ is the \emph{domination number}. When $\mathcal{F}$ is the family of empty graphs, this bound simplifies to $α(G) \le 2γ(G)$ for $K_{1,3}$-free (claw-free) graphs, where $α(G)$ is the \emph{independence number} of $G$. For $d$-regular graphs, this is further refined to the bound $α(G) \le 2\left(\frac{d+1}{d+2}\right)γ(G)$, which is tight for $d \in \{2, 3, 4\}$. Using Ramsey theory, we extend this framework to edge-hereditary graph families, showing that for $K_{1,r}$-free graphs, we have $α_{\mathcal{F}}(G) \le r(K_r, \mathcal{F^*})γ(G)$, where $\mathcal{F^*}$ is the set of graphs not in $\mathcal{F}$. Specializing to $K_q$-free graphs, we show $α_{\mathcal{F}}(G) \le (r(K_q, K_r) - 1)γ(G)$. Finally, for the \emph{$k$-independence number} $α_k(G)$, we prove that if $G$ is $K_{1,r}$-free with order $n$ and minimum degree $δ\ge k+1$, \[ α_k(G) \le \left( \frac{(r-1)(k+1)}{δ- k + (r-1)(k+1)} \right) n, \] and this bound is sharp for all parameters.

Figures (10)

• Figure 1: The $K_{1,4}$-free graph $G = 3K_4 + K_1$ in the proof of Theorem \ref{['thm:general_intro_bound']}
• Figure 2: A set that is both an $\alpha$-set and a $\gamma$-set in a triangle-necklace $F_6$
• Figure 3: The claw-free, subcubic graph $G_{12}$
• Figure 4: An $\alpha$-set in the claw-free, diamond-free cubic graph $H_{2,{\rm cubic}}$
• Figure 5: The claw-free, subcubic graph $G_{20}$

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 1
• Corollary 1
• Theorem 6
• Proposition 1
• Theorem 7