Related papers: Stability number and f-factors in graphs
In this paper, we obtain a sufficient condition for the existence of parity factors in a regular graph in terms of edge-connectivity. Moreover, we also show that our condition is sharp.
$f$-vertex stability number $vs_f(G)=\min\{|X|: X\subseteq V(G) \enspace \text{and} \enspace f(G-X)\neq f(G)\}$, and $f$-edge stability number is defined similarly by setting $X\subseteq E(G)$. In this paper, for multiplicative and mining…
A graph is $t$-tough if the deletion of any set of, say, $m$ vertices from the graph leaves a graph with at most $\frac{m}{t}$ components. In 1973, Chv\'{a}tal suggested the problem of relating toughness to factors in graphs. In 1985,…
We characterize stability of graph C*-algebras by giving five conditions equivalent to their stability. We also show that if G is a graph with no sources, then C*(G) is stable if and only if each vertex in G can be reached by an infinite…
This paper introduces and studies the stability of the strong domination number of a graph, denoted $\operatorname{st}_{\gamma_{st}}(G)$, defined as the minimum number of vertices whose removal changes the strong domination number…
We give a sharp spectral condition for the existence of odd cycles in a graph of given order. We also prove a related stability result.
A numbering $f$ of a graph $G$ of order $n$ is a labeling that assigns distinct elements of the set $\left\{ 1,2,\ldots ,n\right\} $ to the vertices of $G$. The strength $\textrm{str}_{f}\left( G\right)$ of a numbering $f:V\left( G\right)…
A typical result in graph theory says that a graph $G$, satisfying certain conditions, has some property $\cal P$. Once such a theorem is established, it is natural to ask how strongly $G$ satisfies $\cal P$. Can one strengthen the result…
In this paper we obtain the stability theorem for the independence number of $G(n, r, 1)$ graphs. This result was previously stated in the paper of M. Pyaderkin but the proof there was incorrect. We introduce the correct proof of the key…
The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the…
The $\{K_2,C_n\}$-factor of a graph is a spanning subgraph whose each component is either $K_2$ or $C_n$. In this paper, a sufficient condition with regard to tight toughness, isolated toughness and binding number bounds to guarantee the…
The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the…
If P is a hereditary property then we show that, for the existence of a perfect f-factor, P is a sufficient condition for countable graphs and yields a sufficient condition for graphs of size aleph_1. Further we give two examples of a…
The stability method is very useful for obtaining exact solutions of many extremal graph problems. Its key step is to establish the stability property which, roughly speaking, states that any two almost optimal graphs of the same order $n$…
A numbering $f$ of a graph $G$ of order $n$ is a labeling that assigns distinct elements of the set $\left\{ 1,2,\ldots ,n\right\} $ to the vertices of $G$, where each $uv\in E\left( G\right) $ is labeled $f\left( u\right) +f\left( v\right)…
A numbering $f$ of a graph $G$ of order $n$ is a labeling that assigns distinct elements of the set $\{1,2, \ldots, n \}$ to the vertices of $G$. The strength $\mathrm{str}\left(G\right) $ of $G$ is defined by $\mathrm{str}\left( G\right)…
The main result of this paper is an edge-coloured version of Tutte's $f$-factor theorem. We give a necessary and sufficient condition for an edge-coloured graph $G^c$ to have a properly coloured $f$-factor. We state and prove our result in…
For any graph $G$ of order $p$, a bijection $f: V(G)\to [1,p]$ is called a numbering of the graph $G$ of order $p$. The strength $str_f(G)$ of a numbering $f: V(G)\to [1,p]$ of $G$ is defined by $str_f(G) = \max\{f(u)+f(v)\; |\; uv\in…
Let $G$ be a graph. We denote by $e(G)$ and $\rho(G)$ the size and the spectral radius of $G$. A spanning subgraph $F$ of $G$ is called an even factor of $G$ if $d_F(v)\in\{2,4,6,\ldots\}$ for every $v\in V(G)$. Yan and Kano provided a…
Let us say that a convex function f\colon C\to[-\infty,\infty] on a convex set C\subseteq\R is infimum-stable if, for any sequence (f_n) of convex functions f_n\colon C\to[-\infty,\infty] converging to f pointwise, one has \inf_C…