Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2409.17306

Bounds on the Complete Forcing Number of Graphs

Javad B. Ebrahimi·Aref Nemayande·Elahe Tohidi
View original source ↗Read source document ↗Export metadata ↓

Abstract

A forcing set for a perfect matching of a graph is defined as a subset of the edges of that perfect matching such that there exists a unique perfect matching containing it. A complete forcing set for a graph is a subset of its edges, such that it intersects the edges of every perfect matching in a forcing set of that perfect matching. The size of a smallest complete forcing set of a graph is called the complete forcing number of the graph. In this paper, we derive new upper bounds for the complete forcing number of graphs in terms of other graph theoretical parameters such as the degeneracy or the spectral radius of the graph. We show that for graphs with the number of edges more than some constant times the number of vertices, our result outperforms the best known upper bound for the complete forcing number. For the set of edge-transitive graphs, we present a lower bound for the complete forcing number in terms of maximum forcing number. This result in particular is applied to the hypercube graphs and Cartesian powers of even cycles.

Explore related subjects

math.COcs.DM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Javad B. Ebrahimi, Aref Nemayande, Elahe Tohidi. 2024-09-25. Bounds on the Complete Forcing Number of Graphs. https://arxiv.org/abs/2409.17306

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗