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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)

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)

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)

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)

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)

A graph G is said to be pancyclic if G contains cycles of all lengths from 3 to |V (G)|. We show that if G is 4-connected, claw-free, and P 10-free, then G is either pancyclic or it is the line graph of the Pe-tersen graph. This implies that every 4-connected, claw-free, P 9-free graph is pancyclic, which is best possible and extends a result of Gould et al.

1 Introduction and Terminology

A graph G is pancyclic if it contains cycles of each length , 3 ≤ ≤ |V (G)|. The generalized bull B(i, j) is obtained by associating one endpoint of each of the paths P i+1 and P j+1 with distinct vertices of a triangle. Gould, Luczak and Pfender [4] showed that if G is a 3-connected {K 1,3 , B(i, j)}-free graph with i + j = 4 then G is pancyclic. In this… (More)

In [Many disjoint dense subgraphs versus large k-connected subgraphs in large graphs with given edge density, Discrete Math. 309 (2009), 997–1000.], Böhme and Kostochka showed that every large enough graph with sufficient edge density contains either a k-connected subgraph of order at least r or a family of r vertex-disjoint k-connected sub-graphs.… (More)

For a fixed (multi)graph H, a graph G is H-linked if any injection f : V (H) → V (G) can be extended to an H-subdivision in G. The notion of an H-linked graph encompasses several familiar graph classes, including k-linked, k-ordered and k-connected graphs. In this paper, we give two sharp Ore-type degree sum conditions that assure a graph G is H-linked for… (More)