A nonincreasing sequence of nonnegative integers π = (d1, d2, ..., 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 terms of π. For a graph H and n ∈ Z+, σ(H,...

We consider a variation of the classical Turán-type extremal problem as introduced by Erdős, Jacobson and Lehel in [4]. Let π be an n-element graphic sequence and let H be a graph. We wish to determine the smallest even integer m such that any n-term graphic sequence π having degree sum at least m has some realization containing H as a subgraph. Denote this value m by σ(H,n). For an arbitrarily...

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 graphic sequence with sum at least σ(H,...

A nonincreasing sequence n of n nonnegative integers is said to be graphic if it is the degree sequence of a simple graph G of order n and G is called a realization of n. A graph G of order n is said to have property P, if it contains a clique of size k as a subgraph. An n-term graphic sequence n is said to be potentially (res. forcibly)Pk-graphic if it has a realization having (res. all its re...

A bisection of a graph is a balanced bipartite spanning subgraph. Bollobás and Scott conjectured that every graphG has a bisectionH such that degH(v) ≥ bdegG(v)/2c for all vertices v. We prove a degree sequence version of this conjecture: given a graphic sequence π, we show that π has a realization G containing a bisection H where degH(v) ≥ b(degG(v)− 1)/2c for all vertices v. This bound is ver...

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 Fk denote the friendship graph on 2k+1 vertices, that is, the graph of k triangles intersecting in a single vertex. In this paper, for n sufficiently large, σ(...

MOTIVATION Genome projects for many prokaryotic and eukaryotic species have been completed and more new genome projects are being underway currently. The availability of a large number of genomic sequences for researchers creates a need to find graphic tools to study genomes in a perceivable form. The Z curve is one of such tools available for visualizing genomes. The Z curve is a unique three-...

In this paper we consider a variation of the classical Turán-type extremal problems. Let S be an n-term graphical sequence, and σ(S) be the sum of the terms in S. Let H be a graph. The problem is to determine the smallest even l such that any n-term graphical sequence S having σ(S) ≥ l has a realization containing H as a subgraph. Denote this value l by σ(H, n). We show σ(C2m+1, n) = m(2n−m− 1)...