scholarly journals On Fractional Realizations of Graph Degree Sequences

10.37236/3683 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Michael D. Barrus
Keyword(s):  
Degree Sequence ◽  
Convex Polytope ◽  
Simple Graph ◽  
Degree Sequences ◽  
Graph Structure ◽  
Forbidden Subgraphs ◽  
Graph Realization ◽  
Split Graphs

We introduce fractional realizations of a graph degree sequence and a closely associated convex polytope. Simple graph realizations correspond to a subset of the vertices of this polytope; we characterize degree sequences for which each polytope vertex corresponds to a simple graph realization. These include the degree sequences of threshold and pseudo-split graphs, and we characterize their realizations both in terms of forbidden subgraphs and graph structure.

PARTIAL RESULT ON HADWIGER'S CONJECTURE

2010 ◽  
Vol 02 (03) ◽  
pp. 413-423 ◽  
Author(s):  
ZI-XIA SONG
Keyword(s):  
Degree Sequence ◽  
Simple Graph ◽  
Partial Result ◽  
Degree Sequences ◽  
Induced Subgraph ◽  
Graphic Sequence ◽  
Hadwiger's Conjecture ◽  
Hadwiger’S Conjecture

Let D = (d1, d2, …, dn) be a graphic sequence with 0 ≤ d1 ≤ d2 ≤ ⋯ ≤ dn. Any simple graph G with D its degree sequence is called a realization of D. Let R[D] denote the set of all realizations of D. We say that D is H-free if no graph in R[D] contains H as an induced subgraph. In this paper, we prove that Hadwiger's Conjecture is true for graphs whose degree sequences are claw-free or [Formula: see text]-free.

scholarly journals On the Swap-Distances of Different Realizations of a Graphical Degree Sequence

2013 ◽  
Vol 22 (3) ◽  
pp. 366-383 ◽  
Author(s):  
PÉTER L. ERDŐS ◽  
ZOLTÁN KIRÁLY ◽  
ISTVÁN MIKLÓS
Keyword(s):  
Social Networks ◽  
Markov Chain ◽  
Degree Sequence ◽  
Upper Bounds ◽  
Simple Graph ◽  
Related Question ◽  
Degree Sequences ◽  
Uniform Sampling ◽  
Integer Sequence ◽  
The Greedy Algorithm

One of the first graph-theoretical problems to be given serious attention (in the 1950s) was the decision whether a given integer sequence is equal to the degree sequence of a simple graph (orgraphical, for short). One method to solve this problem is the greedy algorithm of Havel and Hakimi, which is based on theswapoperation. Another, closely related question is to find a sequence of swap operations to transform one graphical realization into another of the same degree sequence. This latter problem has received particular attention in the context of rapidly mixing Markov chain approaches to uniform sampling of all possible realizations of a given degree sequence. (This becomes a matter of interest in the context of the study of large social networks, for example.) Previously there were only crude upper bounds on the shortest possible length of such swap sequences between two realizations. In this paper we develop formulae (Gallai-type identities) for theswap-distances of any two realizations of simple undirected or directed degree sequences. These identities considerably improve the known upper bounds on the swap-distances.

scholarly journals Forced Edges and Graph Structure

2019 ◽  
Vol 124 ◽  
Author(s):  
Brian Cloteaux
Keyword(s):  
Degree Sequence ◽  
Degree Sequences ◽  
Graph Structure ◽  
The Relationship

For a degree sequence, we define the set of edges that appear in every labeled realization of that sequence as forced, while the edges that appear in none are define as forbidden. We examine the structure of graphs in which the degree sequences contain either forced or forbidden edges. The results include the determination of the structure of the forced or forbidden edge sets, the relationship between the sizes of forced and forbidden sets for a sequence, and the structural consequences to their realizations. This includes showing that the diameter of every realization of a degree sequence containing forced or forbidden edges is no greater than 3, and that these graphs are maximally edge-connected.

scholarly journals Nonconvexity of the Set of Hypergraph Degree Sequences

10.37236/2719 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Ricky Ini Liu
Keyword(s):  
Convex Polytope ◽  
Simple Graph ◽  
Degree Sequences ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
K 13

It is well known that the set of possible degree sequences for a simple graph on $n$ vertices is the intersection of a lattice and a convex polytope. We show that the set of possible degree sequences for a simple $k$-uniform hypergraph on $n$ vertices is not the intersection of a lattice and a convex polytope for $k \geq 3$ and $n \geq k+13$. We also show an analogous nonconvexity result for the set of degree sequences of $k$-partite $k$-uniform hypergraphs and the generalized notion of $\lambda$-balanced $k$-uniform hypergraphs.

scholarly journals A Simple Havel–Hakimi Type Algorithm to Realize Graphical Degree Sequences of Directed Graphs

10.37236/338 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Péter L. Erdős ◽  
István Miklós ◽  
Zoltán Toroczkai
Keyword(s):  
Greedy Algorithm ◽  
Directed Graphs ◽  
Degree Sequence ◽  
Finite Sequence ◽  
Simple Graph ◽  
Degree Sequences ◽  
Mcmc Algorithm ◽  
Type Algorithm ◽  
Positive Integers ◽  
The Greedy Algorithm

One of the simplest ways to decide whether a given finite sequence of positive integers can arise as the degree sequence of a simple graph is the greedy algorithm of Havel and Hakimi. This note extends their approach to directed graphs. It also studies cases of some simple forbidden edge-sets. Finally, it proves a result which is useful to design an MCMC algorithm to find random realizations of prescribed directed degree sequences.

scholarly journals The Polytope of Degree Partitions

10.37236/1072 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Amitava Bhattacharya ◽  
S. Sivasubramanian ◽  
Murali K. Srinivasan
Keyword(s):  
Convex Hull ◽  
Linear Inequality ◽  
Degree Sequence ◽  
Extreme Points ◽  
Main Tool ◽  
Simple Graph ◽  
Degree Sequences ◽  
Simple Graphs ◽  
Vertex Set ◽  
Partition Polytope

The degree partition of a simple graph is its degree sequence rearranged in weakly decreasing order. The polytope of degree partitions (respectively, degree sequences) is the convex hull of degree partitions (respectively, degree sequences) of all simple graphs on the vertex set $[n]$. The polytope of degree sequences has been very well studied. In this paper we study the polytope of degree partitions. We show that adding the inequalities $x_1\geq x_2 \geq \cdots \geq x_n$ to a linear inequality description of the degree sequence polytope yields a linear inequality description of the degree partition polytope and we show that the extreme points of the degree partition polytope are the $2^{n-1}$ threshold partitions (these are precisely those extreme points of the degree sequence polytope that have weakly decreasing coordinates). We also show that the degree partition polytope has $2^{n-2}(2n-3)$ edges and $(n^2 -3n + 12)/2$ facets, for $n\geq 4$. Our main tool is an averaging transformation on real sequences defined by repeatedly averaging over the ascending runs.

scholarly journals New Results on Degree Sequences of Uniform Hypergraphs

10.37236/3414 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Sarah Behrens ◽  
Catherine Erbes ◽  
Michael Ferrara ◽  
Stephen G. Hartke ◽  
Benjamin Reiniger ◽  
...  
Keyword(s):  
Efficient Algorithm ◽  
Necessary Conditions ◽  
Degree Sequence ◽  
Sufficient Conditions ◽  
Degree Sequences ◽  
Uniform Hypergraph ◽  
Graphic Sequence ◽  
Uniform Hypergraphs ◽  
Open Question

A sequence of nonnegative integers is $k$-graphic if it is the degree sequence of a $k$-uniform hypergraph. The only known characterization of $k$-graphic sequences is due to Dewdney in 1975. As this characterization does not yield an efficient algorithm, it is a fundamental open question to determine a more practical characterization. While several necessary conditions appear in the literature, there are few conditions that imply a sequence is $k$-graphic. In light of this, we present sharp sufficient conditions for $k$-graphicality based on a sequence's length and degree sum.Kocay and Li gave a family of edge exchanges (an extension of 2-switches) that could be used to transform one realization of a 3-graphic sequence into any other realization. We extend their result to $k$-graphic sequences for all $k \geq 3$. Finally we give several applications of edge exchanges in hypergraphs, including generalizing a result of Busch et al. on packing graphic sequences.

scholarly journals Graph classes characterized both by forbidden subgraphs and degree sequences

2007 ◽  
Vol 57 (2) ◽  
pp. 131-148 ◽  
Author(s):  
Michael D. Barrus ◽  
Mohit Kumbhat ◽  
Stephen G. Hartke
Keyword(s):  
Degree Sequences ◽  
Forbidden Subgraphs ◽  
Graph Classes

scholarly journals HAMILTONICITY OF CAMOUFLAGE GRAPHS

2021 ◽  
Vol 5 (10) ◽  
Author(s):  
Sowmiya K
Keyword(s):  
Planar Graphs ◽  
Degree Sequence ◽  
Simple Graph ◽  
Hamilton Path ◽  
Graphic Sequence ◽  
Vertex Transitive ◽  
Connected Vertex

This paper examines the Hamiltonicity of graphs having some hidden behaviours of some other graphs in it. The well-known mathematician Barnette introduced the open conjecture which becomes a theorem by restricting our attention to the class of graphs which is 3-regular, 3- connected, bipartite, planar graphs having odd number of vertices in its partition be proved as a Hamiltonian. Consequently the result proved in this paper stated that “Every connected vertex-transitive simple graph has a Hamilton path” shows a significant improvement over the previous efforts by L.Babai and L.Lovasz who put forth this conjecture. And we characterize a graphic sequence which is forcibly Hamiltonian if every simple graph with degree sequence is Hamiltonian. Thus we discussed about the concealed graphs which are proven to be Hamiltonian.

scholarly journals A variant of Niessen’s problem on degreesequences of graphs

2014 ◽  
Vol Vol. 16 no. 1 (Graph Theory) ◽  
Author(s):  
Jiyun Guo ◽  
Jianhua Yin
Keyword(s):  
Graph Theory ◽  
Discrete Math ◽  
Simple Graph ◽  
The Other ◽  
Degree Sequences ◽  
International Audience

Graph Theory International audience Let (a1,a2,\textellipsis,an) and (b1,b2,\textellipsis,bn) be two sequences of nonnegative integers satisfying the condition that b1>=b2>=...>=bn, ai<= bi for i=1,2,\textellipsis,n and ai+bi>=ai+1+bi+1 for i=1,2,\textellipsis, n-1. In this paper, we give two different conditions, one of which is sufficient and the other one necessary, for the sequences (a1,a2,\textellipsis,an) and (b1,b2,\textellipsis,bn) such that for every (c1,c2,\textellipsis,cn) with ai<=ci<=bi for i=1,2,\textellipsis,n and &#x2211;&limits;i=1n ci=0 (mod 2), there exists a simple graph G with vertices v1,v2,\textellipsis,vn such that dG(vi)=ci for i=1,2,\textellipsis,n. This is a variant of Niessen\textquoterights problem on degree sequences of graphs (Discrete Math., 191 (1998), 247&#x2013;253).

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