Edge-critical Graphs Research Articles

Graphs with large maximum degree containing no edge-critical graphs

We say that a graph is edge-critical if it contains an edge whose deletion reduces its chromatic number. Let F be an edge-critical graph with chromatic number r+2. For sufficiently large n, we determine the maximum number of edges in an n-vertex graph with given maximum degree that does not contain a copy of F. Moreover, the unique extremal graph is a complete (r+1)-partite graph.

A Möbius-type gluing technique for obtaining edge-critical graphs

Using a technique which is inspired by topology, we construct original examples of 3 - and 4 -edge critical graphs. The 3 -critical graphs cover all even orders starting from 26 ; the 4 -critical graphs cover all even orders starting from 20 and all the odd orders. In particular, the 3 -critical graphs are not isomorphic to the graphs provided by Goldberg for disproving the Critical Graph Conjecture. Using the same approach we also revisit the construction of some fundamental critical graphs, such as Goldberg’s infinite family of 3 -critical graphs, Chetwynd’s 4 -critical graph of order 16 and Fiol’s 4 -critical graph of order 18 .

Total Roman domination edge-critical graphs

A total Roman dominating function on a graph $G$ is a function $% f:V(G)\rightarrow \{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ is adjacent to some vertex $u$ with $f(u)=2$, and the subgraph of $G$ induced by the set of all vertices $w$ such that $f(w)>0$ has no isolated vertices. The weight of $f$ is $\Sigma _{v\in V(G)}f(v)$. The total Roman domination number $\gamma _{tR}(G)$ is the minimum weight of a total Roman dominating function on $G$. A graph $G$ is $k$-$\gamma _{tR}$-edge-critical if $\gamma _{tR}(G+e)<\gamma _{tR}(G)=k$ for every edge $e\in E(\overline{G})\neq \emptyset $, and $k$-$\gamma _{tR}$-edge-supercritical if it is $k$-$\gamma _{tR}$-edge-critical and $\gamma _{tR}(G+e)=\gamma _{tR}(G)-2$ for every edge $e\in E(\overline{G})\neq \emptyset $. We present some basic results on $\gamma_{tR}$-edge-critical graphs and characterize certain classes of $\gamma _{tR}$-edge-critical graphs. In addition, we show that, when $k$ is small, there is a connection between $k$-$\gamma _{tR}$-edge-critical graphs and graphs which are critical with respect to the domination and total domination numbers.

Some sharp results on the generalized Turán numbers

For graphs T,H, let ex(n,T,H) denote the maximum number of copies of T in an n-vertex H-free graph. In this paper we prove some sharp results on this generalization of Turán numbers, where our focus is for the graphs T,H satisfying χ(T)<χ(H). This can be dated back to Erdős (1962), where he generalized the celebrated Turán’s theorem by showing that for any r≥m, the Turán graph Tr(n) uniquely attains ex(n,Km,Kr+1). For general graphs H with χ(H)=r+1>m, Alon and Shikhelman (2016) showed that ex(n,Km,H)=rm(nr)m+o(nm). Here we determine this error term o(nm) up to a constant factor. We prove that ex(n,Km,H)=rm(nr)m+biex(n,H)⋅Θ(nm−2), where biex(n,H) is the Turán number of the decomposition family of H. As a special case, we extend Erdős’ result, by showing that Tr(n) uniquely attains ex(n,Km,H) for any edge-critical graph H. We also consider T being non-clique, where even the simplest case seems to be intricate. Following from a more general result, we show that for all s≤t, T2(n) maximizes the number of Ks,t in n-vertex triangle-free graphs if and only if t<s+12+2s+14.

On domination game stable graphs and domination game edge-critical graphs

Let γg(G) be the game domination number of a graph G. For any vertex v∈V(G), we denote by G|v a partially dominated graph G in which the vertex v is already dominated. In this paper, based on the domination game critical graph, two related definitions to the game domination number are introduced as follows. A graph G is domination game stable (or γg-stable for short) if γg(G)=γg(G|v) for any vertex v∈V(G) and is domination game edge-critical (or γg-edge-critical for short) if γg(G−e)>γg(G) for any edge e∈E(G).A γg-stable (γg-edge-critical resp.) graph G is called k-γg-stable (k-γg-edge-critical resp.) if γg(G)=k. We characterize the k-γg-stable graphs with k∈{1,2} and 3-γg-stable trees and show that the Kneser graphs K(n,2) with n≥5 are γg-stable. Also allγg-stable paths and cycles are completely determined, respectively. Moreover, we prove the existence of k-γg-stable graphs for any positive integer k. In addition, all complete r-partite γg-stable and γg-edge-critical graphs are determined, respectively. The characterization of k-γg-edge-critical graphs is given for k∈{1,2} with an example of a3-γg-edge-critical graph. Two sufficient conditions are presented for non-γg-edge-critical graphs. Finally we propose several open related problems.

3-Factor-Criticality in Double Domination Edge Critical Graphs

A vertex subset S of a graph G is a double dominating set of G if $$|N[v]\cap S|\ge 2$$|N[v]?S|?2 for each vertex v of G, where N[v] is the set of the vertex v and vertices adjacent to v. The double domination number of G, denoted by $$\gamma _{\times 2}(G)$$?×2(G), is the cardinality of a smallest double dominating set of G. A graph G is said to be double domination edge critical if $$\gamma _{\times 2}(G+e)<\gamma _{\times 2}(G)$$?×2(G+e)<?×2(G) for any edge $$e \notin E$$e?E. A double domination edge critical graph G with $$\gamma _{\times 2}(G)=k$$?×2(G)=k is called k-$$\gamma _{\times 2}(G)$$?×2(G)-critical. A graph G is r-factor-critical if $$G-S$$G-S has a perfect matching for each set S of r vertices in G. In this paper we show that G is 3-factor-critical if G is a 3-connected claw-free 4-$$\gamma _{\times 2}(G)$$?×2(G)-critical graph of odd order with minimum degree at least 4 except a family of graphs.

Total Domination Edge Critical Graphs with Total Domination Number Three and Many Dominating Pairs

A graph $$G$$G is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter-2-critical graph $$G$$G of order $$n$$n is at most $$\lfloor n^2/4 \rfloor $$?n2/4? and that the extremal graphs are the complete bipartite graphs $$K_{{\lfloor n/2 \rfloor },{\lceil n/2 \rceil }}$$K?n/2?,?n/2?. A graph is $$3_t$$3t-edge-critical, abbreviated $$3_tEC$$3tEC, if its total domination number is 3 and the addition of any edge decreases the total domination number. It is known that proving the Murty---Simon Conjecture is equivalent to proving that the number of edges in a $$3_tEC$$3tEC graph of order $$n$$n is greater than $$\lceil n(n-2)/4 \rceil $$?n(n-2)/4?. We study a family $$\mathcal{F}$$F of $$3_tEC$$3tEC graphs of diameter 2 for which every pair of nonadjacent vertices dominates the graph. We show that the graphs in $$\mathcal{F}$$F are precisely the bull-free $$3_tEC$$3tEC graphs and that the number of edges in such graphs is at least $$\lfloor (n^2 - 4)/4 \rfloor $$?(n2-4)/4?, proving the conjecture for this family. We characterize the extremal graphs, and conjecture that this improved bound is in fact a lower bound for all $$3_tEC$$3tEC graphs of diameter 2. Finally we slightly relax the requirement in the definition of $$\mathcal{F}$$F--instead of requiring that all pairs of nonadjacent vertices dominate to requiring that only most of these pairs dominate--and prove the Murty---Simon equivalent conjecture for these $$3_tEC$$3tEC graphs.

Critical Graphs in Total K Domination

In this paper we study the concept of edge critical graph w. r. t total  K domination. We have derived characterization of edges which are responsible to increase or decrease total  K domination Number of a graph. We have also given examples to illustrate the conditions given in theorems.

Extending Berge’s and Favaron’s results about well-covered graphs

A graph is well-covered if all its maximal independent sets have the same order. For a well-covered graph G of order n(G) without an isolated vertex, Claude Berge (C. Berge, Some common properties for regularizable graphs, edge-critical graphs and b-graphs, Ann. Discrete Math. 12 (1982) 31–44) proved that the independence number α(G) of G is at most n(G)2. The extremal graphs for this result are known as the very well-covered graphs and were characterized by Odile Favaron (O. Favaron, Very well-covered graphs, Discrete Math. 42 (1982) 177–187).We extend these two results in two different ways. First, we study the structure and recognition of the well-covered graphs G without an isolated vertex that have independence number n(G)−k2 for some non-negative integer k. For k=1, we give a complete structural description of these graphs, and for a general but fixed k, we describe a polynomial time recognition algorithm. Second, we relax the condition of well-coveredness and consider graphs G without an isolated vertex for which the independence number α(G) and the independent domination number i(G) satisfy α(G)−i(G)≤k for some non-negative integer k. We prove a suitable version of Berge’s result for these graphs, derive an upper bound on the independence number as a corollary, and discuss its relation to Favaron’s characterization.

Total edge critical graphs with leaves

Let γt(G) denote the total domination number of the graph G. The graph G is said to be total domination edge critical, or simply γt(G)-critical, if γt(G+e)<γt(G) for each e∈E(G¯). We show that any γt(G)-critical graph G with γt(G)≥5 has at most γt(G)−2 leaves, and characterize those γt-critical graphs G having exactly γt(G)−2 leaves. We also constructively establish the existence (with one exception) of h-critical graphs G with k leaves, where k is any nonnegative integer at most h−2. Finally, we characterize the γt-critical unicyclic graphs.

Some results on Roman domination edge critical graphs

A Roman dominating function on a graph G with vertex set V(G) is a function f: V(G) → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function f is the value f(V(G)) = Σu∊v(G)f(u). The Roman domination number, γR(G), of G is the minimum weight of a Roman dominating function on G. In this paper we continue the study of Roman domination edge critical graphs by giving several properties and characterizations for these graphs.

Minimum H-decompositions of graphs: Edge-critical case

For a given graph H let ϕH(n) be the maximum number of parts that are needed to partition the edge set of any graph on n vertices such that every member of the partition is either a single edge or it is isomorphic to H. Pikhurko and Sousa conjectured that ϕH(n)=ex(n,H) for χ(H)⩾3 and all sufficiently large n, where ex(n,H) denotes the maximum size of a graph on n vertices not containing H as a subgraph. In this article, their conjecture is verified for all edge-critical graphs. Furthermore, it is shown that the graphs maximizing ϕH(n) are (χ(H)−1)-partite Turán graphs.

A characterization of locating-total domination edge critical graphs

For a graph G = (V,E) without isolated vertices, a subset D of vertices of V is a total dominating set (TDS) of G if every vertex in V is adjacent to a vertex in D. The total domination number t(G) is the minimum cardinality of a TDS of G. A subset D of V which is a total dominating set, is a locating-total dominating set, or just a LTDS of G, if for any two distinct vertices u and v of V (G) n D, NG(u) D 6 NG(v) D. The locating-total domination number t L(G)

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 critical and stable graphs upon edge removal

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S . The minimum cardinality of a total dominating set of G is the total domination number of G . A graph is total domination edge critical if the removal of any arbitrary edge increases the total domination number. On the other hand, a graph is total domination edge stable if the removal of any arbitrary edge has no effect on the total domination number. In this paper, we characterize total domination edge critical graphs. We also investigate various properties of total domination edge stable graphs.

Edge critical cops and robber

We consider edge critical graphs when playing cops and robber. Specifically, we look at those graphs whose copnumbers change from one to two when any edge is added, deleted, subdivided or contracted. We characterize all such sets, showing that they are empty, trees, all 2-edge-connected graphs and empty, respectively. We also consider those graphs which change from copnumber two to one when any edge is added, and give a characterization in the k-regular case.

MATCHING PROPERTIES IN DOUBLE DOMINATION EDGE CRITICAL GRAPHS

A vertex subset S of a graph G = (V, E) is a double dominating set for G if |N[v]∩S| ≥ 2 for each vertex v ∈ V, where N[v] = {u |uv ∈ E}∪{v}. The double domination number of G, denoted by γ×2(G), is the cardinality of a smallest double dominating set of G. A graph G is said to be double domination edge critical if γ×2(G + e) &lt; γ×2(G) for any edge e ∉ E. A double domination edge critical graph G with γ×2(G) = k is called k - γ×2(G)-critical. In this paper, we first show that G has a perfect matching if G is a connected 3 - γ×2(G)-critical graph of even order. Secondly, we show that G is factor-critical if G is a connected 3 - γ×2(G)-critical graph with odd order and minimum degree at least 2. Finally, we show that G is factor-critical if G is a connected K1,4-free 4 - γ×2(G)-critical graph of odd order with minimum degree at least 2.

Some matching properties in 4-γ×2-critical graphs

A vertex subset S of a graph G = ( V , E ) is a double dominating set for G if | N [ v ] ∩ S | ≥ 2 for each vertex v ∈ V , where N [ v ] = { u | u v ∈ E } ∪ { v } . The double domination number of G , denoted by γ × 2 ( G ) , is the cardinality of a smallest double dominating set of G . A graph G is said to be double domination edge critical if γ × 2 ( G + e ) < γ × 2 ( G ) for any edge e ∉ E . A double domination edge critical graph G with γ × 2 ( G ) = k is called k - γ × 2 ( G ) - critical . In this paper we first show that G has a perfect matching if G is a connected K 1 , 4 -free 4- γ × 2 ( G ) -critical graph of even order ≥6 except a family of graphs. Secondly, we show that G is bicritical if G is a 2-connected claw-free 4- γ × 2 ( G ) -critical graph of even order with minimum degree at least 3. Finally, we show that G is bicritical if G is a 3-connected K 1 , 4 -free 4- γ × 2 ( G ) -critical graph of even order with minimum degree at least 4.

4-chromatic edge critical Grötzsch–Sachs graphs

Let G be a 4-regular plane graph and suppose that G has a cycle decomposition S (i.e., each edge of G is in exactly one cycle of the decomposition) with every pair of adjacent edges on a face always in different cycles of S . Such a graph G arises as a superposition of simple closed curves in the plane with tangencies disallowed. Two 4-chromatic edge critical graphs of order 48 generated by four curves are presented.

Odd cycles and [formula omitted]-cycles in hypergraphs

A Θ -cycle of a hypergraph is a cycle including an edge that contains at least three base points of the cycle. We show that if a hypergraph H = ( V , E ) has no Θ -cycle, and | e | ⩾ 3 , for every edge e ∈ E , then ∑ e ∈ E ( | e | - 1 ) ⩽ 2 | V | - 2 with equality if and only if H is obtained from a hypertree by doubling its edges. This result reminiscent of Berge's and Lovász's similar inequalities implies that 3-uniform hypergraphs with n vertices and n edges have Θ -cycles, and 3-uniform simple hypergraphs with n vertices and n - 1 edges have Θ -cycles. Both results are sharp. Since the presence of a Θ -cycle implies the presence of an odd cycle, both results are sharp for odd cycles as well. However, for linear 3-uniform hypergraphs the thresholds are different for Θ -cycles and for odd cycles. Linear 3-uniform hypergraphs with n vertices and with minimum degree two have Θ -cycles when | E | ⩾ 5 n / 6 - c 1 n and have odd cycles when | E | ⩾ 7 n / 9 - c 2 n and these are sharp results apart from the values of the constants. Most of our proofs use the concept of edge-critical (minimally 2-connected) graphs introduced by Dirac and by Plummer. In fact, the hypergraph results—in disguise—are extremal results for bipartite graphs that have no cycles with chords.