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Fractional forcing number of graphs

Javad B. Ebrahimi, Babak Ghanbari

TL;DR

This work introduces the fractional forcing function, extending the classical forcing number to fractional perfect matchings on the polytope $\mathcal{P}_f(G)$ and proving that this extension is a concave, continuous function that agrees with the integral forcing number on the domain of perfect matchings. The authors establish the theoretical foundations, showing how the fractional forcing number extends the discrete notion and deriving consequences for regular edge-transitive graphs. They demonstrate that the maximum fractional forcing occurs at symmetric, edge-uniform points, enabling nontrivial upper bounds for $F(G)$, with a new bound for hypercube graphs $Q_n$. Additionally, they present a code-based alternative bound and discuss the broader implications of connecting discrete forcing theory with convex analysis. These results provide a framework to translate continuous, geometric insights into bounds on forcing numbers in graph families relevant to chemistry and combinatorics.

Abstract

The notion of forcing sets for perfect matchings was introduced by Harary, Klein, and Živković. The application of this problem in chemistry, as well as its interesting theoretical aspects, made this subject very active. In this work, we introduce the notion of forcing function of fractional perfect matchings, which is continuous analogous to forcing sets defined over the perfect matching polytope of graphs. We show that this object is a continuous and concave function extension of the integral forcing set. Then, we use our results in the continuous world to conclude new bounds and results in the discrete case of forcing sets, for the family of regular edge-transitive graphs. In particular, we derive new upper bounds for the maximum forcing number of hypercube graphs.

Fractional forcing number of graphs

TL;DR

This work introduces the fractional forcing function, extending the classical forcing number to fractional perfect matchings on the polytope and proving that this extension is a concave, continuous function that agrees with the integral forcing number on the domain of perfect matchings. The authors establish the theoretical foundations, showing how the fractional forcing number extends the discrete notion and deriving consequences for regular edge-transitive graphs. They demonstrate that the maximum fractional forcing occurs at symmetric, edge-uniform points, enabling nontrivial upper bounds for , with a new bound for hypercube graphs . Additionally, they present a code-based alternative bound and discuss the broader implications of connecting discrete forcing theory with convex analysis. These results provide a framework to translate continuous, geometric insights into bounds on forcing numbers in graph families relevant to chemistry and combinatorics.

Abstract

The notion of forcing sets for perfect matchings was introduced by Harary, Klein, and Živković. The application of this problem in chemistry, as well as its interesting theoretical aspects, made this subject very active. In this work, we introduce the notion of forcing function of fractional perfect matchings, which is continuous analogous to forcing sets defined over the perfect matching polytope of graphs. We show that this object is a continuous and concave function extension of the integral forcing set. Then, we use our results in the continuous world to conclude new bounds and results in the discrete case of forcing sets, for the family of regular edge-transitive graphs. In particular, we derive new upper bounds for the maximum forcing number of hypercube graphs.

Paper Structure

This paper contains 7 sections, 31 theorems, 39 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Preliminaries
  4. Perfect matching and fractional perfect matching polytope
  5. Main Result
  6. Application
  7. Acknowledgements

Key Result

proposition 2.1

A mapping $f: V\left(C\right) \rightarrow V(G)$ is a homomorphism of cycle $C = (v_1, \dots, v_k)$ to $G$ if and only if $f(v_1), \cdots, f(v_k)$ is a closed walk in $G$.

Figures (2)

  • Figure 1: An example of integral perfect matching
  • Figure 2: The vertices of $Q_4$ are indexed with integer numbers from $0$ to $15$ which are depicted in both decimal and binary representations. The narrow edges represent the blue edges, and the bold edges represent the remaining edges of $Q_4$ (i.e. the red edges).

Theorems & Definitions (41)

  • proposition 2.1: RN200
  • definition 1
  • remark 1
  • Lemma 2.1: Convex Combination
  • definition 2
  • proposition 2.2: RN32(Private communication with Noga Alon)
  • remark 2
  • proposition 2.3: RN250
  • proposition 2.4: RN300
  • proposition 2.5
  • ...and 31 more