Graphs with large maximum forcing number

TL;DR

The paper addresses how the edge density of a graph with a perfect matching governs its forcing properties. It proves the conjectured edge bound $e(G)\ge\frac{n^2}{n-F(G)}$, equivalently $F(G)\le n-\frac{n^2}{e(G)}$, and shows stronger structural consequences in bipartite graphs: if $F(G)\ge n-k$, then any two perfect matchings can be transformed into one another via matching switches along even cycles of length at most $2(k+1)$. It also derives a lower bound on the minimum forcing number $f(G)$ for such graphs, proves continuity-like results for forcing spectra, and characterizes the interval of minimum forcing numbers when $F(G)=n-2$, with several constructions demonstrating sharpness. The results link density, forcing, and switch-based reconfiguration of perfect matchings, and have implications for spectrum structure and the behavior of graphs like hypercubes under forcing operations.

Abstract

For a graph $G$ with order $2n$ and a perfect matching, let $f(G)$ and $F(G)$ denote the minimum and maximum forcing number of $G$ respectively. Then $0\leq f(G)\leq F(G)\leq n-1$. Liu and Zhang [10] ever proposed a conjecture: $e(G)\geq \frac{n^2}{n-F(G)}$, where $e(G)$ denotes the number of edges of $G$. In this paper we confirm this conjecture and obtain $F(G)\leq n-\frac{n^2}{e(G)}$. If $F(G)=n-1$, Liu and Zhang [9] proved that any two perfect matchings of $G$ can be obtained from each other by a series of matching switches along 4-cycles. If $G$ is bipartite and $F(G)\geq n-k$, $1\leq k\leq n-1$, we show that any two perfect matchings of $G$ can be obtained from each other by a series of matching switches along even cycles of length at most $2(k+1)$. Finally, we ask whether $f(G)\geq \lceil\frac{n}{k}\rceil-1$ holds for such bipartite graphs $G$, and give positive answers for the cases $k=1,2$. Further we show all minimum forcing numbers of the bipartite graphs $G$ of order $2n$ and with $F(G)=n-2$ form an integer interval $[\lfloor\frac{n}{2}\rfloor, n-2]$.

Figures (8)

• Figure 1: $M$- and $M'$-alternating cycle $C$ with a chord $v_1u_{1+j}$.
• Figure 2: The form of an $M$-alternating cycle $C'$ in $G'$.
• Figure 3: Constructed process of $M$-alternating cycles in $G'$.
• Figure 4: An $M'$-alternating cycle consisting of $t+1$ chords and $t+1$$M'$-alternating paths.
• Figure 5: A bipartite graph $G_{n,k}$ for Remark \ref{['rmk1']}.

Theorems & Definitions (41)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Corollary 2.2
• Theorem 3.1
• proof
• Corollary 3.2