Results on the edge-connectivity of graphs

It is shown that if G is a graph of order p ≥ 2 such that deg u + deg v ≥ p ? 1 for all pairs u, v of nonadjacent vertices, then the edge-connectivity of G equals the minimum degree of G. Furthermore, if deg u + deg v ≥ p for all pairs u, v of nonadjacent vertices, then either p is even and G is isomorphic to $\text{K}{}_{\mathrm{<mtext>p</mtext><mtext>2</mtext>}}\text{× K}{}_2$ or every minimum cutset of edges of G consists of the collection of edges incident with a vertex of least degree.     

Stability,domination and irredundance in a graph

In a graph G, a set X is called a stable set if any two vertices of X are nonadjacent. A set X is called a dominating set if every vertex of V – X is joined to at least one vertex of X. A set X is called an irredundant set if every vertex of X, not isolated in X, has at least one proper neighbor, that is a vertex of V – X joined to it but to no other vertex of X. Let α′ and α, γ, and Γ, ir and IR, denote respectively the minimum and maximum cardinalities of a maximal stable set, a minimal dominating set, and a maximal irredundant set. It is known that ir ? γ ? α′ ? α ? Γ ? IR and that if G does not contain any induced subgraph isomorphic to K_1,3, then γ = α′. Here we prove that if G contains no induced subgraph isomorphic to K_1,3 or to the graph H of figure 1, then ir = γ = α′. We prove also that if G contains no induced subgraph isomorphic to K_1,3, to H, or to the graph h of figure 3, then Γ = IR. Finally, we improve a result of Bollobas and Cockayne about sufficient conditions for γ = ir in terms of forbidden subgraphs.     

Disjoint independent dominating sets in graphs

A set X of vertices of G is an independent dominating set if no two vertices of X are adjacent and each vertex not in X is adjacent to at least one vertex in X. Independent dominating sets of G are cliques of the complement $\text{G}$ of G and conversely.This work is concerned with the existence of disjoint independent dominating sets in a graph G. A new parameter, the maximum number of disjoint independent dominating sets in G, is studied and the class of graphs whose vertex sets partition into independent dominating sets is investigated.     

On subgraph number independence in trees

For finite graphs F and G, let N_F(G) denote the number of occurrences of F in G, i.e., the number of subgraphs of G which are isomorphic to F. If $\text{F}$ and $\text{G}$ are families of graphs, it is natural to ask then whether or not the quantities N_F(G), F∈$\text{F}$, are linearly independent when G is restricted to $\text{G}$. For example, if $\text{F}$ = {K₁, K₂} (where K_n denotes the complete graph on n vertices) and $\text{F}$ is the family of all (finite) trees, then of course N_K1(T) ? N_K2(T) = 1 for all T∈$\text{F}$. Slightly less trivially, if $\text{F}$ = {S_n: n = 1, 2, 3,…} (where S_n denotes the star on n edges) and $\text{G}$ again is the family of all trees, then Σ_n=1^∞(?1)ⁿ⁺¹N_Sn(T)=1 for all T∈$\text{G}$. It is proved that such a linear dependence can never occur if $\text{F}$ is finite, no F∈$\text{F}$ has an isolated point, and $\text{G}$ contains all trees. This result has important applications in recent work of L. Lovász and one of the authors (Graham and Lovász, to appear).     

Contributions to the theory of domination,independence and irredundance in graphs

A vertex x in a subset X of vertices of an undirected graph is redundant if its closed neighbourhood is contained in the union of closed neighbourhoods of vertices of X?{x}. In the context of a communications network, this means that any vertex which may receive communications from X may also be informed from X?{x}. The lower and upper irredundance numbers ir(G) and IR(G) are respectively the minimum and maximum cardinalities taken over all maximal sets of vertices having no redundancies. The domination number γ(G) and upper domination number Γ(G) are respectively the minimum and maximum cardinalities taken over all minimal dominating sets of G. The independent domination number i(G) and the independence number β(G) are respectively the minimum and maximum cardinalities taken over all maximal independent sets of vertices of G.A variety of inequalities involving these quantities are established and sufficient conditions for the equality of the three upper parameters are given. In particular a conjecture of Hoyler and Cockayne [9], namely $\text{i+β?2p + 2δ - 2}\text{2pδ}$, is proved.     

On the automorphism group of a tree

It is shown that H = Γ(T)_v is normal in G = Γ(T_v) for any tree T and any vertex v, if and only if, for all vertices u in the neighborhood N of v, the set of images of u under G is either contained in N or has precisely the vertex u in common with N and every vertex in the set of images is fixed by H. Further, if S is the smallest normal subgroup of G containing H then $\text{G}\text{S}$ is the direct product of the wreath products of various symmetric groups around groups of order 1 or 2. The degrees of the symmetric groups involved depend on the numbers of isomorphic components of T_v and the structure of such components.     

Convex hull of the edges of a graph and near bipartite graphs

First we characterize the convex hull of the edges of a graph, edges viewed as the characteristic function of the hereditary closure of some subset of the 2-elements set of a finite set X. This characterization becomes more simple for a class of graphs that we call near bipartite, NBP in short. This class is then characterized as the class of graphs such that ?x?X, G_X\r(x), the induced subgraph of the complementary of the neighbourhood of x, is bipartite. We made a partial study of this class, whose interest is justified by the constatation that the following classes are strictly include: $\text{L(G)}$ the edge complementary of the line graph of G. NBP, $\text{K}{}_{13}$-free graphs.     

Some tree-star Ramsey Numbers

The Ramsey Number r(G₁, G₂) is the least integer N such that for every graph G with N vertices, either G has the graph G₁ as a subgraph or $\text{G}$, the complement of G, has the graph G₂ as a subgraph.In this paper we embed the paths P_m in a much larger class $\text{T}$ of trees and then show how some evaluations by T. D. Parsons of Ramsey numbers r(P_m, K_1,n), where K_1,n is the star of degree n, are also valid for r(T_m, K_1,n) where T_m ∈ $\text{T}$.     

On the maximum number of diagonals of a circuit in a graph

For a graph G, let ?(G) denote the maximum number k such that G contains a circuit with k diagonals.Theorem. For any graph G with minimum valency$\text{n? 3, ?(G) ?}\text{1}\text{2}\text{(n+1)(n-2)}$.If the equality holds and G is connected, then either G is isomorphic to K_n+1 or G is separable and each of its terminal blocks is isomorphic to K_n+1, or K_n+1 with one edge subdivided.     

Graph equations for line graphs,total graphs,middle graphs and quasi-total graphs

Let G be a graph with vertex-set V(G) and edge-set X(G). Let L(G) and T(G) denote the line graph and total graph of G. The middle graph M(G) of G is an intersection graph Ω(F) on the vertex-set V(G) of any graph G. Let F = V′(G) ∪ X(G) where V′(G) indicates the family of all one-point subsets of the set V(G), then $\text{M(G) = Ω(F)}$.The quasi-total graph P(G) of G is a graph with vertex-set V(G)∪X(G) and two vertices are adjacent if and only if they correspond to two non-adjacent vertices of G or to two adjacent edges of G or to a vertex and an edge incident to it in G.In this paper we solve graph equations $\text{L(G) ? P(H);}\text{L(G)}\text{? P(H); P(G) ? T(H);}\text{P(G)}\text{? T(H); M(G) ? P(H);}\text{M(G)}\text{? P(H)}$.     

A lower bound for the independence number of a planar graph

A subset of the vertices of a graph is independent if no two vertices in the subset are adjacent. The independence number α(G) is the maximum number of vertices in an independent set. We prove that if G is a planar graph on N vertices then $\text{α(G)/N ?}\text{2}\text{9}$.     

On intersections of interval graphs

If one can associate with each vertex of a graph an interval of a line, so that two intervals intersect just when the corresponding vertices are joined by an edge, then one speaks of an interval graph.It is shown that any graph on v vertices is the intersection (“product”) of at most [$\text{1}\text{2}$v] interval graphs on the same vertex set.For v=2k, k factors are necessary for, and only for, the complete k-partite graph K_2,2,…,2.Some results for the hypergraph generalization of this question are also obtained.     

Graphical regular representations of free products of groups

If X is a Cayley graph of a group G possessing a normal subgroup N, then there is a quotient graph of X which is a Cayley graph of $\text{G}\text{N}$. With the aid of this result, it is shown that the free product of at least two and at most countably many groups, each of which is at most countably generated, admits a graphical regular representation.     

Every 3-connected,locally connected,claw-free graph is Hamilton-connected

A graph G is locally connected if the subgraph induced by the neighbourhood of each vertex is connected. We prove that a locally connected graph G of order p ≥ 4, containing no induced subgraph isomorphic to K_1,3, is Hamilton-connected if and only if G is 3-connected. © 1996 John Wiley & Sons, Inc.     

Quadripartite version of the Hajnal-Szemerédi theorem

Let G be a quadripartite graph with N vertices in each vertex class and each vertex is adjacent to at least vertices in each of the other classes. There exists an N₀ such that, if N?N₀, then G contains a subgraph that consists of N vertex-disjoint copies of K₄.     

Edge-deletable IM-extendable graphs with minimum number of edges

A graph G is induced matching extendable, shortly IM-extendable, if every induced matching of G is included in a perfect matching of G. For a nonnegative integer k, a graph G is called a k-edge-deletable IM-extendable graph, if, for every F⊆E(G) with |F|=k, G−F is IM-extendable. In this paper, we characterize the k-edge-deletable IM-extendable graphs with minimum number of edges. We show that, for a positive integer k, if G is ak-edge-deletable IM-extendable graph on 2n vertices, then |E(G)|≥(k+2)n; furthermore, the equality holds if and only if either G≅K_k+2,k+2, or k=4r−2 for some integer r≥3 and G≅C₅[N_2r], where N_2r is the empty graph on 2r vertices and C₅[N_2r] is the graph obtained from C₅ by replacing each vertex with a graph isomorphic to N_2r.     

Matchings in 3-vertex-critical graphs: The odd case

A subset of vertices D of a graph G is a dominating set for G if every vertex of G not in D is adjacent to one in D. The cardinality of any smallest dominating set in G is denoted by γ(G) and called the domination number of G. Graph G is said to be γ-vertex-critical if γ(G-v)<γ(G), for every vertex v in G. A graph G is said to be factor-critical if G-v has a perfect matching for every choice of v∈V(G).In this paper, we present two main results about 3-vertex-critical graphs of odd order. First we show that any such graph with positive minimum degree and at least 11 vertices which has no induced subgraph isomorphic to the bipartite graph K_1,5 must contain a near-perfect matching. Secondly, we show that any such graph with minimum degree at least three which has no induced subgraph isomorphic to the bipartite graph K_1,4 must be factor-critical. We then show that these results are best possible in several senses and close with a conjecture.     

On n-hamiltonian graphs

A graph G of order p ? 3 is called n-hamiltonian, 0 ? n ? p ? 3, if the removal of any m vertices, 0 ? m ? n, results in a hamiltonian graph. A graph G of order p ? 3 is defined to be n-hamiltonian, ?p ? n ? 1, if there exists ?n or fewer pairwise disjoint paths in G which collectively span G. Various conditions in terms of n and the degrees of the vertices of a graph are shown to be sufficient for the graph to be n-hamiltonian for all possible values of n. It is also shown that if G is a graph of order p ? 3 and $\text{K}\text{(G) ? β(G) + n (?p ? n ? p ? 3)}$, then G is n-hamiltonian.     

Graphical t-wise balanced designs

A pair (X, $\text{B}$) will be a t-wise balanced design (tBD) of type t?(v, K, λ) if $\text{B}\text{= (B}{}_{\mathrm{i}}\text{: i ? I)}$ is a family of subsets of X, called blocks, such that: (i) $\text{|X| = v ?}\text{N}$, where $\text{N}$ is the set of positive integers; (ii) $\text{1?t?|B}{}_{\mathrm{i}}\text{|?K?}\text{N}$, for every i ? I; and (iii) if T ? X, |T| = t, then there are λ ? $\text{N}$ indices i ? I where T ? B_i. Throughout this paper we make three restrictions on our tBD's: (1) there are no repeated blocks, i.e. $\text{B}$ will be a set of subsets of X; (2) t ? K or there are no blocks of size t; and (3) $\text{P}{}_{\mathrm{k}}\text{(X)?}\text{B}$ or $\text{B}$ does not contain all k-subsets of X for any t<k?v. Note then that X ? $\text{B}$. Also, if we give the parameters of a specific tBD, then we will choose a minimal K.We focus on the $\text{t?((}\text{p}\text{2}\text{), K, λ)}$ designs with the symmetric group S_p as automorphism group, i.e. X will be the set of $\text{v = (}\text{p}\text{2}\text{)}$ labelled edges of the undirected complete graph K_p and if B ? $\text{B}$ then all subgraphs of K_p isomorphic to B are also in $\text{B}$. Call such tBD's ‘graphical tBD's’. We determine all graphical tBD's with λ = 1 or 2 which will include one with parameters 4?(15,{5,7},1).     

Minimum number of edges in graphs that are both P2- and Pi-connected

A simple graph with n vertices is called P_i-connected if any two distinct vertices are connected by an elementary path of length i. In this paper, lower bounds of the number of edges in graphs that are both P₂- and P_i-connected are obtained. Namely if $\text{i?}\text{1}\text{2}\text{(n+1)}$, then |E(G)|?((4i?5)/(2i?2))(n?1), and if $\text{i >}\text{1}\text{2}\text{(n+ 1)}$, then |E(G)|?2(n?1) apart from one exeptional graph. Furthermore, extremal graphs are determined in the former.