Given a graph G = (V,E), a vertex subset S is called t-stable (or t-dependent) if the subgraph G[S] induced on S has maximum degree at most t. The t-stability number of G is the maximum order of a t-stable set in G. We investigate the typical values that this parameter takes on a random graph on n vertices and edge probability equal to p. For any fixed 0 < p < 1 and fixed non-negative integer t, we show that, with probability tending to 1 as n grows, the t-stability number takes on at most two values which we identify as functions of t, p and n. The main tool we use is an asymptotic expression for the expected number of t-stable sets of order k. We derive this expression by performing a precise count of the number of graphs on k vertices that have maximum degree at most k. Using the above results, we also obtain asymptotic bounds on the t-improper chromatic number of a random graph (this is the generalisation of the chromatic number, where we partition of the vertex set of the graph into t-stable sets).
Understanding which system structure can sustain stable dynamics is a fundamental step in the design and analysis of large scale dynamical systems. Towards this goal, we investigate here the structural stability of systems with a random structure. As is usually done, we describe the system's structure through a graph describing interactions between parts of the system, and we call a graph stable if it describe a structurally stable system, i.e., a system which can sustain stable dynamics. We then consider two Erdos-Renyi random graph models, and we obtain for each the probability that a graph …
We extend the classical stability theorem of Erdos and Simonovits in two directions: first, we allow the order of the forbidden graph to grow as log of order of the host graph, and second, our extremal condition is on the spectral radius of the host graph.
The following sharpening of Turán's theorem is proved. Let $T_{n,p}$ denote the complete $p$--partite graph of order $n$ having the maximum number of edges. If $G$ is an $n$-vertex $K_{p+1}$-free graph with $e(T_{n,p})-t$ edges then there exists an (at most) $p$-chromatic subgraph $H_0$ such that $e(H_0)\geq e(G)-t$. Using this result we present a concise, contemporary proof (i.e., one applying Szemerédi's regularity lemma) for the classical stability result of Simonovits.
A graph with convex quadratic stability number is a graph for which the stability number is determined by solving a convex quadratic program. Since the very beginning, where a convex quadratic programming upper bound on the stability number was introduced, a necessary and sufficient condition for this upper bound be attained was deduced. The recognition of graphs with convex quadratic stability number has been deeply studied with several consequences from continuous and combinatorial point of view. This survey starts with an exposition of some extensions of the classical Motzkin-Straus approac…
A $p$-improper interval graph is an interval graph that has an interval representation in which no interval contains more than $p$ other intervals. A critical $p$-improper interval graph is $p-1$ improper when any vertex is removed. In this paper we investigate the spectrum of impropriety of critical $p$-improper interval graphs upon the removal of a single vertex, which is informally known as the stability of the graph.
A pair of graphs $(Γ,Σ)$ is said to be stable if the full automorphism group of $Γ\timesΣ$ is isomorphic to the product of the full automorphism groups of $Γ$ and $Σ$ and unstable otherwise, where $Γ\timesΣ$ is the direct product of $Γ$ and $Σ$. In this paper, we reduce the study of the stability of any pair of regular graphs $(Γ,Σ)$ with coprime valencies and vertex-transitive $Σ$ to that of $(Γ,K_2)$. Since the latter is well studied in the literature, this enables us to determine the stability of any pair of regular graphs $(Γ,Σ)$ with coprime valencies in the case when $Σ$ is vertex-transi…
For an arbitrary invariant $ρ(G)$ of a graph $G$, the $ρ-$vertex stability number $vs_ρ(G)$ is the minimum number of vertices of $G$ whose removal results in a graph $H\subseteq G$ with $ρ(H)\neq ρ(G)$ or with $E(H)=\varnothing$. In this paper, first we give some general lower and upper bounds for the $ρ$-vertex stability number, and then study the edge chromatic stability number of graphs, $vs_{χ^{\prime}}(G)$, where $χ^{\prime}=χ^{\prime}(G)$ is edge chromatic number (chromatic index) of $G$. We prove some general results for this parameter and determine $vs_{χ^{\prime}}(G)$ for specific cla…
The chromatic edge-stability number ${\rm es}_χ(G)$ of a graph $G$ is the minimum number of edges whose removal results in a spanning subgraph $G'$ with $χ(G')=χ(G)-1$. Edge-stability critical graphs are introduced as the graphs $G$ with the property that ${\rm es}_χ(G-e) < {\rm es}_χ(G)$ holds for every edge $e\in E(G)$. If $G$ is an edge-stability critical graph with $χ(G)=k$ and ${\rm es}_χ(G)=\ell$, then $G$ is $(k,\ell)$-critical. Graphs which are $(3,2)$-critical and contain at most four odd cycles are classified. It is also proved that the problem of deciding whether a graph $G$ has $χ(…
The {\em chromatic edge-stability number} $es_χ(G)$ of a graph $G$ is the minimum number of edges whose removal results in a spanning subgraph with the chromatic number smaller than that of $G$. A graph $G$ is called {\em $(3,2)$-critical} if $χ(G)=3$, $es_χ(G)=2$ and for any edge $e\in E(G)$, $es_χ(G-e)<es_χ(G)$. In this paper, we characterize $(3,2)$-critical graphs which contain at least five odd cycles. This answers a question proposed by Brešar, Klavžar and Movarraei in [Critical graphs for the chromatic edge-stability number, {\it Discrete Math.} {\bf 343}(2020) 111845].
Popielarz, Sahasrabudhe and Snyder in 2018 proved that maximal $K_{r+1}$-free graphs with $(1-\frac{1}{r})\frac{n^2}{2}-o(n^{\frac{r+1}{r}})$ edges contain a complete $r$-partite subgraph on $n-o(n)$ vertices. This was very recently extended to odd cycles in place of $K_3$ by Wang, Wang, Yang and Yuan. We further extend it to some other 3-chromatic graphs, and obtain some other stability results along the way.
The chromatic vertex (resp.\ edge) stability number ${\rm vs}_χ(G)$ (resp.\ ${\rm es}_χ(G)$) of a graph $G$ is the minimum number of vertices (resp.\ edges) whose deletion results in a graph $H$ with $χ(H)=χ(G)-1$. In the main result it is proved that if $G$ is a graph with $χ(G) \in \{ Δ(G), Δ(G)+1 \}$, then ${\rm vs}_χ(G) = {\rm ivs}_χ(G)$, where ${\rm ivs}_χ(G)$ is the independent chromatic vertex stability number. The result need not hold for graphs $G$ with $χ(G) \le \frac{Δ(G)+1}{2}$. It is proved that if $χ(G) > \frac{Δ(G)}{2}+1$, then ${\rm vs}_χ(G) = {\rm es}_χ(G)$. A Nordhaus-Gaddum-…
For any positive integer $k$, we show that every maximal $C_{2k+1}$-free graph with at least $n^2/4-o(n^{3/2})$ edges contains an induced complete bipartite subgraph on $(1-o(1))n$ vertices. We also show that this is best possible.
The (independent) chromatic vertex stability ($\ivs(G)$) $\vs(G)$ is the minimum size of (independent) set $S\subseteq V(G)$ such that $χ(G-S)=χ(G)-1$. In this paper we construct infinitely many graphs $G$ with $Δ(G)=4$, $χ(G)=3$, $\ivs(G)=3$ and $\vs(G)=2$, which gives a partial negative answer to a problem posed in \cite{ABKM}.
The classical stability theorem of Erdős and Simonovits states that, for any fixed graph with chromatic number $k+1 \ge 3$, the following holds: every $n$-vertex graph that is $H$-free and has within $o(n^2)$ of the maximal possible number of edges can be made into the $k$-partite Turán graph by adding and deleting $o(n^2)$ edges. In this paper, we prove sharper quantitative results for graphs $H$ with a critical edge, both for the Erdős-Simonovits Theorem (distance to the Turán graph) and for the closely related question of how close an $H$-free graph is to being $k$-partite. In many cases, t…
In the special case of graphs G of independence number a(G)=3 without induced chordless cycles C7 it is shown that exists connected dominating set D of vertices with number of vertices n(D)<=4. Using the concept of connected dominating sets, we defined a new invariant h(G) that does not exceed the number of Hadwiger. For the considered graphs it is shown that h(G)>=n(G)/4.
Given graphs $H$ and $F$, the generalized Turán number $\mathrm{ex}(n,H,F)$ is the largest number of copies of $H$ in $n$-vertex $F$-free graphs. Stability refers to the usual phenomenon that if an $n$-vertex $F$-free graph $G$ contains almost $\mathrm{ex}(n,H,F)$ copies of $H$, than $G$ is in some sense similar to some extremal graph. We obtain new stability results for generalized Turán problems and derive several new exact results.
Both the Simonovits stability theorem and the Nikiforov spectral stability theorem are powerful tools for solving exact values of Turán numbers in extremal graph theory. Recently, Füredi [J. Combin. Theory Ser. B 115 (2015)] provided a concise and contemporary proof of the Simonovits stability theorem. In this note, we present a unified treatment for some extremal graph problems, including short proofs of Nikiforov's spectral stability theorem and the clique stability theorem proved recently by Ma and Qiu [European J. Combin. 84 (2020)]. Moreover, some spectral extremal problems related to the…
The dynamical properties of finite dynamical systems (FDSs) have been investigated in the context of coding theoretic problems, such as network coding, and in the context of hat games, such as the guessing game and Winkler's hat game. The instability of an FDS is the minimum Hamming distance between a state and its image under the FDS, while the stability is the minimum of the reciprocal of the Hamming distance; they are both directly related to Winkler's hat game. In this paper, we study the value of the (in)stability of FDSs with prescribed interaction graphs. The first main contribution of …
A graph $X$ is said to be unstable if the direct product $X\times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is non-trivially unstable if it is unstable, connected, non-bipartite, and distinct vertices have distinct sets of neighbours. In this paper, we prove two sufficient conditions for stability of graphs in which every edge lies on a triangle, revising an incorrect claim of Surowski and filling in some gaps in the proof of another one. We also consider triangle-free graphs, and prove that there…
We start up the study of the stability of general graph pairs. This notion is a generalization of the concept of the stability of graphs. We say that a pair of graphs $(Γ,Σ)$ is stable if $Aut(Γ\timesΣ) \cong Aut(Γ)\times Aut(Σ)$ and unstable otherwise, where $Γ\timesΣ$ is the direct product of $Γ$ and $Σ$. An unstable graph pair $(Γ,Σ)$ is said to be a nontrivially unstable graph pair if $Γ$ and $Σ$ are connected coprime graphs, at least one of them is non-bipartite, and each of them has the property that different vertices have distinct neighbourhoods. We obtain necessary conditions for a pa…
For $r \geq 2$, we show that every maximal $K_{r+1}$-free graph $G$ on $n$ vertices with $(1-\frac{1}{r})\frac{n^2}{2}-o(n^{\frac{r+1}{r}})$ edges contains a complete $r$-partite subgraph on $(1 - o(1))n$ vertices. We also show that this is best possible. This result answers a question of Tyomkyn and Uzzell.
