Induced subgraphs of graphs with large deficiency

TL;DR

The paper studies $def(G)$ for connected graphs with large deficiency under a bounded local independence number $α_l(G)$, focusing on bone-structured subgraphs and bone-admitting families $\mathcal{G}(A)$. It proves an upper bound $def(G)\le m(n-3)(n-2)^{(m-3)/2}+1$ for odd $m\ge 3$ when $A$ satisfies a separation condition, with a sharper bound $def(G)\le 2n-5$ when $m=3$, and introduces the Levelling-Matching algorithm to construct large matchings that certify these bounds. The work also characterizes extremal deficiency-critical graphs in special cases, identifying constructions like $S_{n-1}^p$ and $T_{n-2}^p$ that attain the bounds, thereby extending hole-based results to bone configurations. Overall, it provides asymptotically optimal and tight bounds for a broad class of bone-avoiding graphs and frames open questions about parity patterns in the maximum deficiency.

Abstract

The deficiency of a graph $G$, denoted by $\kd(G)$, is the number of vertices not saturated by a maximum matching. A bone $B_i$ is the tree obtained by attaching two pendent edges to each of the end vertices of a path $P_{i}$. The local independence number of $G$, denoted by $α_l(G)$, is defines as the maximum integer $t$ such that $G$ contains an induced star $K_{1,t}$. Motivated by the seminal works of Scott and Seymour~(2016), Chudnovsky et al. (2017, 2020) on finding special types of holes in graphs with large chromatic number and bounded clique number, we establish an analog result by finding special types of bones in graphs with large deficiency and bounded local independence number. Fujita et al. (2006) proved that $\kd(G)\le n-2$ if $G$ is a connected graph with $α_l(G)<n$ and containing no bones. We further establish exact extremal deficiency bounds for connected graphs with bounded local independence number that exclude specific bone configurations. An algorithm that constructs large matchings and establishes an upper bound on the deficiency is also provided.

Figures (2)

• Figure 1: Graphs $B\!S_n^p, S_n^p,T_n^p,E_{m,n}^p$ and $E_{m,n}^{p+}$. All edges exist between the two end of the thick blue line. The red vertex $v$ is the vertex added to $E_{m,n}^{p+}$ from $E_{m,n}^p$.
• Figure 2: $F(a_1,a_2,a_3)$: the dashed line represents path, the blue triangle in $L_0\cup L_{a_1}\cup L_{a_2}$ means the $K_3$ got by $Y$-$\Delta$ operation.

Theorems & Definitions (67)

• Theorem 1.1
• Theorem 1.2: jprls
• Theorem 1.3: fklops
• Theorem 1.4
• proof
• Corollary 1.6
• proof
• Theorem 1.8
• Theorem 1.9
• Lemma 2.1