Vertex Critical Graphs Research Articles

Vertex-critical graphs far from edge-criticality

Abstract Let $r$ be any positive integer. We prove that for every sufficiently large $k$ there exists a $k$ -chromatic vertex-critical graph $G$ such that $\chi (G-R)=k$ for every set $R \subseteq E(G)$ with $|R|\le r$ . This partially solves a problem posed by Erdős in 1985, who asked whether the above statement holds for $k \ge 4$ .

Graphs with Girth $2\ell+1$ and Without Longer Odd Holes that Contain an Odd $K_4$-Subdivision

We say that a graph $G$ has an odd $K_4$-subdivision if some subgraph of $G$ is isomorphic to a $K_4$-subdivision and whose faces are all odd holes of $G$. For a number $\ell\geq 2$, let $\mathcal{G}_{\ell}$ denote the family of graphs which have girth $2\ell+1$ and have no odd hole with length greater than $2\ell+1$. Wu, Xu and Xu conjectured that every graph in $\bigcup_{\ell\geq2}\mathcal{G}_{\ell}$ is $3$-colorable. Recently, Chudnovsky et al. and Wu et al., respectively, proved that every graph in $\mathcal{G}_2$ and $\mathcal{G}_3$ is $3$-colorable. In this paper, we prove that no $4$-vertex-critical graph in $\bigcup_{\ell\geq5}\mathcal{G}_{\ell}$ has an odd $K_4$-subdivision. Using this result, Chen proved that all graphs in $\bigcup_{\ell\geq5}\mathcal{G}_{\ell}$ are $3$-colorable.

The existence of a graph whose vertex set can be partitioned into a fixed number of strong domination-critical vertex-sets

<abstract><p>Let $ \gamma(G) $ denote the domination number of a graph $ G $. A vertex $ v\in V(G) $ is called a critical vertex of $ G $ if $ \gamma(G-v) = \gamma(G)-1 $. A graph is called vertex-critical if its every vertex is critical. In this paper, we correspondingly introduce two such definitions: (i) A set $ S\subseteq V(G) $ is called a strong critical vertex-set of $ G $ if $ \gamma(G-S) = \gamma(G)-|S| $; (ii) A graph $ G $ is called strong $ l $-vertex-set-critical if $ V(G) $ can be partitioned into $ l $ strong critical vertex-sets of $ G $. Therefrom, we give some properties of strong $ l $-vertex-set-critical graphs by extending the previous results of vertex-critical graphs. As the core work, we study on the existence of this class of graphs and prove that there exists a strong $ l $-vertex-set-critical connected graph if and only if $ l\notin\{2, 3, 5\} $.</p></abstract>

Formula omitted]-packing chromatic vertex-critical graphs

For a non-decreasing sequence of positive integers S=(s1,s2,…), the S-packing chromatic numberχS(G) of G is the smallest integer k such that the vertex set of G can be partitioned into sets Xi, i∈[k], where vertices in Xi are pairwise at distance greater than si. In this paper we introduce S-packing chromatic vertex-critical graphs, χS-critical for short, as the graphs in which χS(G−u)<χS(G) for every u∈V(G). This extends the earlier concept of the packing chromatic vertex-critical graphs. We show that if G is χS-critical, then the set {χS(G)−χS(G−u):u∈V(G)} can be almost arbitrary. If G is χS-critical and χS(G)=k (k∈N), then G is called k-χS-critical. We characterize 3-χS-critical graphs and partially characterize 4-χS-critical graphs when s1>1. We also deal with k-χS-criticality of trees and caterpillars.

Critical concept for double Roman domination in graphs

A double Roman dominating function (DRDF) on a graph [Formula: see text] is a function [Formula: see text] such that (i) every vertex [Formula: see text] with [Formula: see text] is adjacent to at least two vertices assigned a [Formula: see text] or to at least one vertex assigned a [Formula: see text] and (ii) every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] with [Formula: see text] The weight of a DRDF is the sum of its function values over all vertices. The double Roman domination number [Formula: see text] equals the minimum weight of a DRDF on [Formula: see text] The concept of criticality with respect to various operations on graphs has been studied for several domination parameters. In this paper, we study the concept of criticality for double Roman domination in graphs. In addition, we characterize double Roman domination edge super critical graphs and we will give several characterizations for double Roman domination vertex (edge) critical graphs.

Independent Roman Domination Stable and Vertex-Critical Graphs

A Roman dominating function (RDF) on a graph $G$ is a function $f: V(G) \rightarrow \{0, 1, 2\}$ for which every vertex assigned 0 is adjacent to a vertex assigned 2. The weight of an RDF is the value $\omega (f) = \sum _{u \in V(G)}f(u)$ . The minimum weight of an RDF on a graph $G$ is called the Roman domination number of $G$ . An RDF $f$ is called an independent Roman dominating function (IRDF) if the set $\{v\in V\mid f(v)\ge 1\}$ is an independent set. The minimum weight of an IRDF on a graph $G$ is called the independent Roman domination number of $G$ and is denoted by $i_{R}(G)$ . A graph $G$ is independent Roman domination stable if the independent Roman domination number of $G$ does not change under removal of any vertex. A graph $G$ is said to be independent Roman domination vertex critical or $i_{R}$ - vertex critical , if for any vertex $v$ in $G$ , $i_{R}(G-v) . In this paper, we characterize independent Roman domination stable trees and we establish upper bounds on the order of independent Roman stable graphs. Also, we investigate the properties of $i_{R}$ - vertex critical graphs. In particular, we present some families of $i_{R}$ -vertex critical graphs and we characterize $i_{R}$ -vertex critical block graphs.

On the Diameter of Total Domination Vertex-Critical Graphs

In this paper, we consider various types of domination vertex critical graphs, including total domination vertex critical graphs, independent domination vertex critical graphs and connected domination vertex critical graphs. We provide upper bounds on the diameter of them, two of which are sharp.

Coloring a graph with [formula omitted] colors: Conjectures equivalent to the Borodin–Kostochka conjecture that appear weaker

Borodin and Kostochka conjectured that every graph G with maximum degree Δ≥9 satisfies χ≤max{ω,Δ−1}. We carry out an in-depth study of minimum counterexamples to the Borodin–Kostochka conjecture. Our main tool is the identification of graph joins that are f-choosable, where f(v)≔d(v)−1 for each vertex v. Such a join cannot be an induced subgraph of a vertex critical graph with χ=Δ, so we have a wealth of structural information about minimum counterexamples to the Borodin–Kostochka conjecture.Our main result proves that certain conjectures that are prima facie weaker than the Borodin–Kostochka conjecture are in fact equivalent to it. One such equivalent conjecture is the following: Any graph with χ=Δ=9 contains K3∨K6¯ as a subgraph.

Partitioning and coloring graphs with degree constraints

We prove that if G is a vertex-critical graph with χ(G)≥Δ(G)+1−p≥4 for some p∈N and ω(H(G))≤χ(G)+1p+1−2, then G=Kχ(G) or G=O5. Here H(G) is the subgraph of G induced on the vertices of degree at least χ(G). This simplifies the proofs and improves the results in the paper of Kostochka, Rabern and Stiebitz (2012) [8].

The equivariant topology of stable Kneser graphs

The stable Kneser graph S G n , k , n ⩾ 1 , k ⩾ 0 , introduced by Schrijver (1978) [19], is a vertex critical graph with chromatic number k + 2 , its vertices are certain subsets of a set of cardinality m = 2 n + k . Björner and de Longueville (2003) [5] have shown that its box complex is homotopy equivalent to a sphere, Hom ( K 2 , S G n , k ) ≃ S k . The dihedral group D 2 m acts canonically on S G n , k , the group C 2 with 2 elements acts on K 2 . We almost determine the ( C 2 × D 2 m ) -homotopy type of Hom ( K 2 , S G n , k ) and use this to prove the following results. The graphs S G 2 s , 4 are homotopy test graphs, i.e. for every graph H and r ⩾ 0 such that Hom ( S G 2 s , 4 , H ) is ( r − 1 ) -connected, the chromatic number χ ( H ) is at least r + 6 . If k ∉ { 0 , 1 , 2 , 4 , 8 } and n ⩾ N ( k ) then S G n , k is not a homotopy test graph, i.e. there are a graph G and an r ⩾ 1 such that Hom ( S G n , k , G ) is ( r − 1 ) -connected and χ ( G ) < r + k + 2 .

Δ-Critical graphs with small high vertex cliques

We prove that K χ ( G ) is the only vertex critical graph G with χ ( G ) ⩾ Δ ( G ) ⩾ 6 and ω ( H ( G ) ) ⩽ ⌊ Δ ( G ) 2 ⌋ − 2 . Here H ( G ) is the subgraph of G induced on the vertices of degree at least χ ( G ) . Setting ω ( H ( G ) ) = 1 proves a conjecture of Kierstead and Kostochka.

The equivariant topology of stable Kneser graphs

Schrijver introduced the stable Kneser graph $SG_{n,k}, n \geq 1, k \geq 0$. This graph is a vertex critical graph with chromatic number $k+2$, its vertices are certain subsets of a set of cardinality $m=2n+k$. Björner and de Longueville have shown that its box complex is homotopy equivalent to a sphere, $\mathrm{Hom}(K_2,SG_{n,k}) \simeq \mathbb{S}^k$. The dihedral group $D_{2m}$ acts canonically on $SG_{n,k}$. We study the $D_{2m}$ action on $\mathrm{Hom}(K_2,SG_{n,k})$ and define a corresponding orthogonal action on $\mathbb{R}^{k+1} \supset \mathbb{S}^k$. We establish a close equivariant relationship between the graphs $SG_{n,k}$ and Borsuk graphs of the $k$-sphere and use this together with calculations in the $\mathbb{Z}_2$-cohomology ring of $D_{2m}$ to tell which stable Kneser graphs are test graphs in the sense of Babson and Kozlov. The graphs $SG_{2s,4}$ are test graphs, i.e. for every graph $H$ and $r \geq 0$ such that $\mathrm{Hom}(SG_{2s,4},H)$ is $(r-1)$-connected, the chromatic number $\chi (H)$ is at least $r+6$. On the other hand, if $k \notin \{0,1,2,4,8\}$ and $n \geq N(k)$ then $SG_{n,k}$ is not a homotopy test graph, i.e. there are a graph $G$ and an $r \geq 1$ such that $\mathrm{Hom}(SG_{n,k}, G)$ is $(r-1)$-connected and $\chi (G) &lt; r+k+2$. The latter result also depends on a new necessary criterion for being a test graph, which involves the automorphism group of the graph. Schrijver a défini le graphe de Kneser stable $SG_{n,k}$, avec $n \geq 1$ et $k \geq 0$. Le graphe $SG_{n,k}$ est un graphe critique (par rapport aux sommets) de nombre chromatique $k+2$, dont les sommets correspondent à certains sous-ensembles d'un ensemble de cardinalité $m=2n+k$. Björner et de Longueville ont démontré que son complexe de boîtes et la sphère sont homotopiquement équivalents, c'est-à-dire $\mathrm{Hom}(K_2,SG_{n,k}) \simeq \mathbb{S}^k$. Le groupe diédral $D_{2m}$ agit sur $SG_{n,k}$ canoniquement. Nous étudions l'action de $D_{2m}$ sur $\mathrm{Hom}(K_2,SG_{n,k})$ et nous définissons une action orthogonale correspondante sur $\mathbb{R}^{k+1} \supset \mathbb{S}^k$. Par ailleurs, nous fournissons une relation équivariante étroite entre les graphes $SG_{n,k}$ et les graphes de Borsuk de la sphère de dimension $k$. Utilisant cette relation et certains calculs dans l'anneau de cohomologie de $D_{2m}$ sur $\mathbb{Z}_2$, nous décrivons quels graphes de Kneser stables sont des graphes de tests selon la notion de Babson et Kozlov. Les graphes $SG_{2s,4}$ sont des graphes de tests, c'est-à-dire que pour tout $H$ et $r \geq 0$ tels que $\mathrm{Hom}(SG_{2s,4},H)$ est $(r-1)$-connexe, le nombre chromatique $\chi (H)$ est au moins $r+6$. D'autre part, si $k \notin \{0,1,2,4,8\}$ et $n \geq N(k)$, alors $SG_{n,k}$ n'est pas un graphe de tests d'homologie: il existe un graphe $G$ et un entier $r \geq 1$ tels que $\mathrm{Hom}(SG_{n,k}, G)$ est $(r-1)$-connexe et $\chi (G) &lt; r+k+2$. Ce dernier résultat dépend d'un nouveau critère nécessaire pour être un graphe de tests, qui implique le groupe d'automorphismes du graphe.

Perfect Matchings in Total Domination Critical Graphs

A graph is total domination edge-critical if the addition of any edge decreases the total domination number, while a graph with minimum degree at least two is total domination vertex-critical if the removal of any vertex decreases the total domination number. A 3 t EC graph is a total domination edge-critical graph with total domination number 3 and a 3 t VC graph is a total domination vertex-critical graph with total domination number 3. A graph G is factor-critical if G − v has a perfect matching for every vertex v in G. In this paper, we show that every 3 t EC graph of even order has a perfect matching, while every 3 t EC graph of odd order with no cut-vertex is factor-critical. We also show that every 3 t VC graph of even order that is K 1,7-free has a perfect matching, while every 3 t VC graph of odd order that is K 1,6-free is factor-critical. We show that these results are tight in the sense that there exist 3 t VC graphs of even order with no perfect matching that are K 1,8-free and 3 t VC graphs of odd order that are K 1,7-free but not factor-critical.

Total domination dot-critical graphs

A graph G with no isolated vertex is total domination vertex-critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G − v is less than the total domination number of G . A graph is total domination dot-critical if contracting any edge decreases the total domination number. In this paper, we study total domination dot-critical graphs. We present several properties of these graphs. We show that the total domination dot-critical graphs include the total domination vertex-critical graphs.

On the existence problem of the total domination vertex critical graphs

The existence problem of the total domination vertex critical graphs has been studied in a series of articles. We first settle the existence problem with respect to the parities of the total domination number m and the maximum degree Δ : for even m except m = 4 , there is no m - γ t -critical graph regardless of the parity of Δ ; for m = 4 or odd m ≥ 3 and for even Δ , an m - γ t -critical graph exists if and only if Δ ≥ 2 ⌊ m − 1 2 ⌋ ; for m = 4 or odd m ≥ 3 and for odd Δ , if Δ ≥ 2 ⌊ m − 1 2 ⌋ + 7 , then m - γ t -critical graphs exist, if Δ < 2 ⌊ m − 1 2 ⌋ , then m - γ t -critical graphs do not exist. The only remaining open cases are Δ = 2 ⌊ m − 1 2 ⌋ + k , k = 1 , 3 , 5 . Second, we study these remaining open cases when m = 4 or odd m ≥ 9 . As the previously known result for m = 3 , we also show that for Δ ( G ) = 3 , 5 , 7 , there is no 4 - γ t -critical graph of order Δ ( G ) + 4 . On the contrary, it is shown that for odd m ≥ 9 there exists an m - γ t -critical graph for all Δ ≥ m − 1 .

Matching Properties in Total Domination Vertex Critical Graphs

A vertex subset S of a graph G = (V,E) is a total dominating set if every vertex of G is adjacent to some vertex in S. The total domination number of G, denoted by γ t (G), is the minimum cardinality of a total dominating set of G. A graph G with no isolated vertex is said to be total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, γ t (G−v) < γ t (G). A total domination vertex critical graph G is called k-γ t -critical if γ t (G) = k. In this paper we first show that every 3-γ t -critical graph G of even order has a perfect matching if it is K 1,5-free. Secondly, we show that every 3-γ t -critical graph G of odd order is factor-critical if it is K 1,5-free. Finally, we show that G has a perfect matching if G is a K 1,4-free 4-γ t (G)-critical graph of even order and G is factor-critical if G is a K 1,4-free 4-γ t (G)-critical graph of odd order.

A note on -vertex critical graphs

A locating-total dominating set of a graph G = (V(G), E(G)) with no isolated vertex is a set S ⊆ V(G) such that every vertex of V(G) is adjacent to a vertex of S and for every pair of distinct vertices u and v in V(G) − S, N(u) ∩ S = N(v) ∩ S. Let be the minimum cardinality of a locating-total dominating set of G. A graph G is said to be locating-total domination vertex critical if for every vertex w, G − w has no isolated vertex and . In this note, we characterize locating-total domination vertex critical Unicycle graphs.

On total domination vertex critical graphs of high connectivity

A graph is called γ -critical if the removal of any vertex from the graph decreases the domination number, while a graph with no isolated vertex is γ t -critical if the removal of any vertex that is not adjacent to a vertex of degree 1 decreases the total domination number. A γ t -critical graph that has total domination number k , is called k - γ t -critical. In this paper, we introduce a class of k - γ t -critical graphs of high connectivity for each integer k ≥ 3 . In particular, we provide a partial answer to the question “Which graphs are γ -critical and γ t -critical or one but not the other?” posed in a recent work [W. Goddard, T.W. Haynes, M.A. Henning, L.C. van der Merwe, The diameter of total domination vertex critical graphs, Discrete Math. 286 (2004) 255–261].

The diameter of paired-domination vertex critical graphs

In this paper we continue the study of paired-domination in graphs introduced by Haynes and Slater (Networks 32 (1998), 199–206). A paired-dominating set of a graph G with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G, denoted by γ pr(G), is the minimum cardinality of a paired-dominating set of G. The graph G is paired-domination vertex critical if for every vertex v of G that is not adjacent to a vertex of degree one, γ pr(G − v) < γ pr(G). We characterize the connected graphs with minimum degree one that are paired-domination vertex critical and we obtain sharp bounds on their maximum diameter. We provide an example which shows that the maximum diameter of a paired-domination vertex critical graph is at least 3/2 (γ pr(G) − 2). For γ pr(G) ⩽ 8, we show that this lower bound is precisely the maximum diameter of a paired-domination vertex critical graph.

Paired Domination Vertex Critical Graphs

Let γpr (G) denote the paired domination number of graph G. A graph G with no isolated vertex is paired domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, γpr (G – v) < γpr (G). We call these graphs γpr -critical. In this paper, we present a method of constructing γpr -critical graphs from smaller ones. Moreover, we show that the diameter of a γpr -critical graph is at most $$\frac{3}{2}(\gamma_{pr} (G)-2)$$and the upper bound is sharp, which answers a question proposed by Henning and Mynhardt [The diameter of paired-domination vertex critical graphs, Czechoslovak Math. J., to appear].