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Let G be a graph and F be a family of graphs. We say that G is F-saturated if G does not contain a copy of any member of F , but for any pair of nonadjacent vertices x and y in G, G + xy contains a copy of some H ∈ F. A great deal of study has been devoted to the maximum and the minimum number of edges in an F-saturated graph. Little is known, however,(More)
For any simple graph H, let σ(H, n) be the minimum m so that for any realizable degree sequence π = (d1, d2,. .. , dn) with sum of degrees at least m, there exists an n-vertex graph G witnessing π that contains H as a weak subgraph. Let F k denote the friendship graph on 2k + 1 vertices, that is, the graph of k triangles intersecting in a single vertex. In(More)
Let ω0(G) denote the number of odd components of a graph G. The deficiency of G is defined as def (G) = max X⊆V (G) (ω0(G − X) − |X|), and this equals the number of vertices unmatched by any maximum matching of G. A subset X ⊆ V (G) is called a Tutte set (or barrier set) of G if def (G) = ω0(G − X) − |X|, and an extreme set if def (G − X) = def (G) + |X|.(More)
A sequence of nonnegative integers π = (d 1 , d 2 , ..., d n) is graphic if there is a (simple) graph G with degree sequence π. In this case, G is said to realize or be a realization of π. Degree sequence results in the literature generally fall into two classes: forcible problems, in which all realizations of a graphic sequence must have a given property,(More)
An integer sequence π is said to be graphic if it is the degree sequence of some simple graph G. In this case we say that G is a realization of π. Given a graph H, and a graphic sequence π we say that π is potentially H-graphic if there is some realization of π that contains H as a subgraph. We define σ(H, n) to be the minimum even integer such that every(More)
An n-tuple π (not necessarily monotone) is graphic if there is a simple graph G with vertex set {v 1 ,. .. , v n } in which the degree of v i is the ith entry of π. Graphic n-tuples (d (2) n) pack if there are edge-disjoint n-vertex graphs G 1 and G 2 such that d G 1 (v i) = d (1) i and d G 2 (v i) = d (2) i for all i. We prove that graphic n-tuples π 1 and(More)
For a fixed multigraph H , possibly containing loops, with V (H) = {h 1 ,. .. , h k }, we say a graph G is H-linked if for every choice of k vertices v 1 ,. .. , v k in G, there exists a subdivision of H in G such that v i represents h i (for all i). This notion clearly generalizes the concept of k-linked graphs (as well as other properties). In this paper(More)
A graph G is H-saturated if G does not contain H as a subgraph but for any nonadjacent vertices u and v, G + uv contains H as a subgraph. The parameter sat(H, n) is the minimum number of edges in an H-saturated graph of order n. In this paper, we determine sat(H, n) for sufficiently large n when H is a union of cliques of the same order, an arbitrary union(More)
We consider an extremal problem for graphs as introduced by Erd˝ os, Jacobson and Lehel in [7]. Let π be an n-element graphic sequence. Let H be a graph. We wish to determine the smallest m such that any n-term graphic sequence π whose terms sum to at least m has some realization containing H as a subgraph. Denote this value m by σ(H, n). For an arbitrarily(More)
A nonincreasing sequence of nonnegative integers π = (d 1 , d 2 , ..., dn) is graphic if there is a (simple) graph G of order n having degree sequence π. In this case, G is said to realize π. For a given graph H, a graphic sequence π is potentially H-graphic if there is some realization of π containing H as a (weak) subgraph. Let σ(π) denote the sum of the(More)