S.B. Rao conjectured in 1971 that graphic degree sequences are well quasi ordered by a relation defined in terms of the induced subgraph relation[6]. In 2008, M. Chudnovsky and P. Seymour proved this long standing Rao’s Conjecture by giving structure theorems for graphic degree sequences[1]. In this paper, we prove and use a fairly simple semigroup lemma to give a short proof of the bounded deg...

Designing networks in which every processor has a given number of connections often leads to graphic degree sequence realization models. A nonincreasing sequence d = (d1, d2, . . . , dn) is graphic if there is a simple graphGwith degree sequence d. The spanning tree packing number of graphG, denoted by τ(G), is themaximumnumber of edge-disjoint spanning trees in G. The arboricity of graph G, de...

In recent years, more and more researchers presented their graphic representations for protein sequence. Here, we present a new 2D graphic representation of protein sequence based on four physicochemical properties of 20 amino acids. By this graphic representation we define a distance calculation formula to quantificationally calculate the similarity degree of different protein sequences. Then ...

Abstract A nonnegative integer sequence is k -graphic if it the degree of a -uniform simple hypergraph. The problem deciding whether given $$\pi $$ π 3-graphic has recently been proved to be NP-complete, after years studies. Thus, acquires primary relevance detect classes sequences whose graphicality can test...

A nonnegative integer sequence is k-graphic if it the degree of a k-uniform simple hypergraph. The problem deciding whether given \(\pi \) admits 3-uniform hypergraph has recently been proved to be NP-complete, after long years research. Thus, helpful find which classes instances are polynomially solvable in order restrict NP-hard core and design algorithms for real-life applications. Several n...

A sequence S is potentially Kp,1,1 graphical if it has a realization containing a Kp,1,1 as a subgraph, where Kp,1,1 is a complete 3-partite graph with partition sizes p, 1, 1. Let σ(Kp,1,1, n) denote the smallest degree sum such that every n-term graphical sequence S with σ(S) ≥ σ(Kp,1,1, n) is potentially Kp,1,1 graphical. In this paper, we prove that σ(Kp,1,1, n) ≥ 2[((p + 1)(n − 1) + 2)/2] ...

We consider an extremal problem for graphs as introduced by Erdős, 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 nterm 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 chosen H, we give a construction that yi...

Gould, Jacobson and Lehel (Combinatorics, Graph Theory and Algorithms, Vol.I (1999) 451–460) considered a variation of the classical Turán-type extremal problems as follows: for any simple graph H, determine the smallest even integer σ(H, n) such that every n-term graphic sequence π = (d1, d2, . . . , dn) with term sum σ(π) = d1 + d2 + · · ·+ dn ≥ σ(H, n) has a realization G containing H as a s...