A Note on the Matching Polytope of a Graph

The matching polytope of a graph G, denoted by M(G), is the convex hull of the set of the incidence vectors of the matchings G. The graph G(M(G)), whose vertices and edges are the vertices and edges of M(G), is the skeleton of the matching polytope of G. In this paper, for an arbitrary graph, we prove that the minimum degree of G(M(G)) is equal to the number of edges of G, generalizing a known result for trees. From this, we identify the vertices of the skeleton with the minimum degree and we prove that the union of stars and triangles characterizes regular skeletons of the matching polytopes of graphs.

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A CHARACTERIZATION OF PM-COMPACT CLAW-FREE CUBIC GRAPHS

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830914500256 ◽

2014 ◽

Vol 06 (02) ◽

pp. 1450025 ◽

Cited By ~ 2

Author(s):

XIUMEI WANG ◽

WEIPING SHANG ◽

YIXUN LIN ◽

MARCELO H. CARVALHO

Keyword(s):

Convex Hull ◽

Perfect Matching ◽

Perfect Matchings ◽

Cubic Graphs ◽

Matching Polytopes ◽

Matching Polytope

The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings in G. This paper characterizes claw-free cubic graphs whose 1-skeleton graphs of perfect matching polytopes have diameter 1.

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On Komlós’ tiling theorem in random graphs

Combinatorics Probability Computing ◽

10.1017/s0963548319000129 ◽

2019 ◽

Vol 29 (1) ◽

pp. 113-127

Author(s):

Rajko Nenadov ◽

Nemanja Škorić

Keyword(s):

Chromatic Number ◽

Random Graph ◽

Random Graphs ◽

Upper Bound ◽

Minimum Degree ◽

Arbitrary Graph ◽

Spanning Subgraph ◽

Delta G ◽

Image Position ◽

Vertex Disjoint

AbstractGiven graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if $p \ge C{n^{ - 1/{m_2}(H)}}$ , then asymptotically almost surely every spanning subgraph G of the random graph 𝒢(n, p) with minimum degree at least $\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}}(H)}} + \gamma )np$ contains an H-tiling that covers all but at most γn vertices. Here, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlós, and m2(H) is the 2-density of H. We show that this theorem can be bootstrapped to obtain an H-tiling covering all but at most $\gamma {(C/p)^{{m_2}(H)}}$ vertices, which is strictly smaller when $p \ge C{n^{ - 1/{m_2}(H)}}$ . In the case where H = K3, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph H we give an upper bound on p for which some leftover is unavoidable and a bound on the size of a largest H -tiling for p below this value.

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Tilings in Randomly Perturbed Dense Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548318000366 ◽

2018 ◽

Vol 28 (2) ◽

pp. 159-176 ◽

Cited By ~ 12

Author(s):

JÓZSEF BALOGH ◽

ANDREW TREGLOWN ◽

ADAM ZSOLT WAGNER

Keyword(s):

High Probability ◽

Minimum Degree ◽

Graph Model ◽

Regularity Lemma ◽

Arbitrary Graph ◽

Random Graph Model ◽

Dense Graphs ◽

Szemeredi's Regularity Lemma ◽

Vertex Disjoint ◽

Special Case

A perfect H-tiling in a graph G is a collection of vertex-disjoint copies of a graph H in G that together cover all the vertices in G. In this paper we investigate perfect H-tilings in a random graph model introduced by Bohman, Frieze and Martin [6] in which one starts with a dense graph and then adds m random edges to it. Specifically, for any fixed graph H, we determine the number of random edges required to add to an arbitrary graph of linear minimum degree in order to ensure the resulting graph contains a perfect H-tiling with high probability. Our proof utilizes Szemerédi's Regularity Lemma [29] as well as a special case of a result of Komlós [18] concerning almost perfect H-tilings in dense graphs.

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Paths and Cycles in Random Subgraphs of Graphs with Large Minimum Degree

The Electronic Journal of Combinatorics ◽

10.37236/6761 ◽

2018 ◽

Vol 25 (2) ◽

Cited By ~ 1

Author(s):

Stefan Ehard ◽

Felix Joos

Keyword(s):

Random Graph ◽

Minimum Degree ◽

Graph Model ◽

Arbitrary Graph ◽

Random Graph Model ◽

Longest Path ◽

Random Subgraphs

For a graph $G$ and $p\in [0,1]$, let $G_p$ arise from $G$ by deleting every edge mutually independently with probability $1-p$. The random graph model $(K_n)_p$ is certainly the most investigated random graph model and also known as the $G(n,p)$-model. We show that several results concerning the length of the longest path/cycle naturally translate to $G_p$ if $G$ is an arbitrary graph of minimum degree at least $n-1$.For a constant $c>0$ and $p=\frac{c}{n}$, we show that asymptotically almost surely the length of the longest path in $G_p$ is at least $(1-(1+\epsilon(c))ce^{-c})n$ for some function $\epsilon(c)\to 0$ as $c\to \infty$, and the length of the longest cycle is a least $(1-O(c^{- \frac{1}{5}}))n$. The first result is asymptotically best-possible. This extends several known results on the length of the longest path/cycle of a random graph in the $G(n,p)$-model to the random graph model $G_p$ where $G$ is a graph of minimum degree at least $n-1$.

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Convex hull of chain-coded blob

IEE Proceedings E Computers and Digital Techniques ◽

10.1049/ip-e.1989.0072 ◽

1989 ◽

Vol 136 (6) ◽

pp. 530

Author(s):

G.R. Wilson ◽

B.G. Batchelor

Keyword(s):

Convex Hull

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The Order of Monochromatic Subgraphs with a Given Minimum Degree

The Electronic Journal of Combinatorics ◽

10.37236/1725 ◽

2003 ◽

Vol 10 (1) ◽

Author(s):

Yair Caro ◽

Raphael Yuster

Keyword(s):

Positive Integer ◽

Minimum Degree

Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order at least $t$. Let $f_G(d)=0$ in case there is a $2$-coloring of the edges of $G$ with no such monochromatic subgraph. Let $f(n,k,d)$ denote the minimum of $f_G(d)$ where $G$ ranges over all graphs with $n$ vertices and minimum degree at least $k$. In this paper we establish $f(n,k,d)$ whenever $k$ or $n-k$ are fixed, and $n$ is sufficiently large. We also consider the case where more than two colors are allowed.

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