Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1809.08460

Some notes on the signed bad number in bipartite graphs

Doost Ali Mojdeh·Babak Samadi
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we deal with the signed bad number and the negative decision number of graphs. We show that two upper bounds concerning these two parameters for bipartite graphs in papers [Discrete Math. Algorithms Appl. 1 (2011), 33--41] and [Australas. J. Combin. 41 (2008), 263--272] are not true as they stand. We correct them by presenting more general bounds for triangle-free graphs by using the classic theorem of Mantel from the extremal graph theory and characterize all triangle-free graphs attaining these bounds.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Doost Ali Mojdeh, Babak Samadi. 2018-09-22. Some notes on the signed bad number in bipartite graphs. https://doi.org/10.1142/s1793830920920019

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

KEEP EXPLORING

Related papers

arXiv2026

Matching Complexes of Outerplanar Graphs

An outerplanar graph is a planar graph that has a planar drawing with all vertices on the unbounded face. The matching complex of a graph is the simplicial complex whose faces are subsets of disjoint edges of the graph. In this paper we prove that the matching complexes of outerplanar graphs are contractible or homotopy equivalent to a wedge of spheres. This extends known results about trees and polygonal line tilings.

math.CO↗
arXiv2026

Perfect matchings and $A_α$-spectral radius in 1-binding graphs

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For $α\in[0,1)$, we use $A_α(G)$ and $ρ_α(G)$ to denote the $A_α$-matrix and the $A_α$-spectral radius of $G$, respectively. The binding number $\mbox{bind}(G)$ of $G$ is defined by $\mbox{bind}(G)=\min\left\{\frac{|N_G(X)|}{|X|}:\emptyset\neq X\subseteq V(G),N_G(X)\neq V(G)\right\}$. If $\mbox{bind}(G)\geq1$, then $G$ is called 1-binding. A perfect matching in $G$ is a set of nonadjacent edges covering every vertex of $G$. Tutte proved that a graph $G$ of even order has a perfect matching if and only if $o(G-S)\leq|S|$ holds for every $S\subseteq V(G)$ [W. Tutte, The factorization of linear graphs, J. Lond. Math. Soc. 22 (1947) 107--111]. In this paper, we use Tutte's result to prove that a connected 1-binding graph $G$ of even order $n$ with $n\geq n(α)$ has a perfect matching unless $G=K_1\vee(K_{n-5}\cup K_3\cup K_1)$ if $ρ_α(G)\geqρ_α(K_1\vee(K_{n-5}\cup K_3\cup K_1))$, where $n(α)$ is defined as follows: $n(α)=\max\{18,\frac{2+8α}{1-2α}\}$ if $α\in[0,\frac{1}{2})$, and $n(α)=18$ if $α=\frac{1}{2}$.

math.CO↗
arXiv2026

Turán number of a matching and a Berge triangle

For a fixed graph $G$, an $r$-uniform hypergraph is said to contain a Berge-$G$ if there exists a bijection $f\colon E(G)\to E(\mathcal{H})$ for some subhypergraph $\mathcal{H}$ such that $e\subseteq f(e)$ for every $e\in E(G)$. Motivated by Alon and Frankl's study of Turán problems under bounded matching constraints, we investigate the maximum number of edges in $r$-uniform Berge-$K_3$-free hypergraphs with matching number at most~$s$. We determine the exact Turán numbers for the cases $r=3$ and $r=4$. For $r=3$ and $n \geq 3 s$, we prove that every $n$-vertex Berge- $K_3$-free 3-graph with matching number $s$ has at most $s(n-2 s)$ edges, and we characterize the unique extremal hypergraph attaining equality. For $r=4$ and $n \geq 4 s$, the maximum number of edges is $s\lfloor(n-2 s) / 2\rfloor$, except for the exceptional case $s=1$ and $n \equiv 1(\bmod 4)$, in which the bound is $(n-1) / 2$. As a corollary, our results recover the classical theorem of Győri on Berge-$K_3$-free hypergraphs.

math.CO↗