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Trees maximizing the number of almost-perfect matchings

Stijn Cambie, Bradley McCoy, Gunjan Sharma, Stephan Wagner, Corrine Yap

TL;DR

This work investigates extremal counts of almost-perfect matchings and strong almost-perfect matchings in trees, deriving sharp bounds that depend on the parity of the order $n$ and completely characterizing extremal structures (odd $n$: at most $\frac{n+1}{2}$ APMs achieved by a $1$-subdivision of a tree of order $\frac{n+1}{2}$; even $n$: at most $\binom{\frac{n}{2}+1}{2}$ APMs, with $P_n$ extremal for $n>4$, and related small-n exceptions). It uses subdivision arguments, degree-sequence majorization, and inductive schemes to prove tight maxima and minima for APMs, SAPMs, and maximal matchings, and it connects these results to the weighted Hosoya index with degree-based weights $Z_\phi(T)$. The authors show that stars minimize many weighted indices while various spider- and broom-like trees maximize them depending on the weight function, highlighting how local degree patterns influence global matching counts. These results advance the combinatorial understanding of near-perfect matchings in trees and provide a framework for exploring weighted enumeration in graph invariants with practical implications in chemistry and network design.

Abstract

We characterize the extremal trees that maximize the number of almost-perfect matchings, which are matchings covering all but one or two vertices, and those that maximize the number of strong almost-perfect matchings, which are matchings missing only one or two leaves. We also determine the trees that minimize the number of maximal matchings. We apply these results to extremal problems on the weighted Hosoya index for several choices of vertex-degree-based weight function.

Trees maximizing the number of almost-perfect matchings

TL;DR

This work investigates extremal counts of almost-perfect matchings and strong almost-perfect matchings in trees, deriving sharp bounds that depend on the parity of the order and completely characterizing extremal structures (odd : at most APMs achieved by a -subdivision of a tree of order ; even : at most APMs, with extremal for , and related small-n exceptions). It uses subdivision arguments, degree-sequence majorization, and inductive schemes to prove tight maxima and minima for APMs, SAPMs, and maximal matchings, and it connects these results to the weighted Hosoya index with degree-based weights . The authors show that stars minimize many weighted indices while various spider- and broom-like trees maximize them depending on the weight function, highlighting how local degree patterns influence global matching counts. These results advance the combinatorial understanding of near-perfect matchings in trees and provide a framework for exploring weighted enumeration in graph invariants with practical implications in chemistry and network design.

Abstract

We characterize the extremal trees that maximize the number of almost-perfect matchings, which are matchings covering all but one or two vertices, and those that maximize the number of strong almost-perfect matchings, which are matchings missing only one or two leaves. We also determine the trees that minimize the number of maximal matchings. We apply these results to extremal problems on the weighted Hosoya index for several choices of vertex-degree-based weight function.
Paper Structure (10 sections, 21 theorems, 26 equations, 12 figures)

This paper contains 10 sections, 21 theorems, 26 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maximum number of matchings of a fixed size
  4. Maximum number of strong almost-perfect matchings
  5. Minimum number of maximal matchings
  6. Extremal values of the weighted Hosoya index
  7. Further Directions
  8. Maximum number of strong $k$-almost-perfect matchings for $k\ge 3$
  9. Weighted Hosoya index
  10. Acknowledgments

Key Result

Theorem 1

If $n$ is odd, a tree $T$ of order $n$ has at most $\frac{n+1}{2}$ almost-perfect matchings. Equality holds if and only if $T$ is a $1$-subdivision of a tree of order $\frac{n+1}{2}.$

Figures (12)

  • Figure 1: An odd spider $\mathop{\mathrm{Sp}}\nolimits_{11}$
  • Figure 2: An even spider $\mathop{\mathrm{Sp}}\nolimits_{12}$
  • Figure 4: A (balanced) double broom $\mathop{\mathrm{DB}}\nolimits_{4,5}$
  • Figure 5: A wide spider $W_{20}$
  • Figure 7: An even special spider $\mathop{\mathrm{SS}}\nolimits_{12}$
  • ...and 7 more figures

Theorems & Definitions (56)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 5
  • Lemma 6
  • proof
  • Lemma 7: Karamata's inequality, Kar32
  • Theorem \ref{thr:odd_apm}
  • proof
  • ...and 46 more