scholarly journals An Entropy Proof of the Kahn-Lovász Theorem

10.37236/497 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Jonathan Cutler ◽  
A. J. Radcliffe
Keyword(s):  
Bipartite Graph ◽  
Upper Bound ◽  
Degree Sequence ◽  
Perfect Matchings ◽  
Entropy Methods ◽  
General Graphs

Brègman gave a best possible upper bound for the number of perfect matchings in a balanced bipartite graph in terms of its degree sequence. Recently Kahn and Lovász extended Brègman's theorem to general graphs. In this paper, we use entropy methods to give a new proof of the Kahn-Lovász theorem. Our methods build on Radhakrishnan's use of entropy to prove Brègman's theorem.

scholarly journals Face-Width of Pfaffian Braces and Polyhex Graphs on Surfaces

10.37236/3540 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Dong Ye ◽  
Heping Zhang
Keyword(s):  
Bipartite Graph ◽  
Polynomial Time ◽  
Perfect Matching ◽  
Cartesian Product ◽  
Perfect Matchings ◽  
Face Width ◽  
Graphs On Surfaces ◽  
Positive Genus ◽  
Pfaffian Graph

A graph $G$ with a perfect matching is Pfaffian if it admits an orientation $D$ such that every central cycle $C$ (i.e. $C$ is of even size and $G-V(C)$ has a perfect matching) has an odd number of edges oriented in either direction of the cycle. It is known that the number of perfect matchings of a Pfaffian graph can be computed in polynomial time. In this paper, we show that every embedding of a Pfaffian brace (i.e. 2-extendable bipartite graph)  on a surface with a positive genus has face-width at most 3.  Further, we study Pfaffian cubic braces and obtain a characterization of Pfaffian polyhex graphs: a polyhex graph is Pfaffian if and only if it is either non-bipartite or isomorphic to the cube, or the Heawood graph, or the Cartesian product $C_k\times K_2$ for even integers $k\ge 6$.

Rainbow Perfect Matchings in Complete Bipartite Graphs: Existence and Counting

2013 ◽  
Vol 22 (5) ◽  
pp. 783-799 ◽  
Author(s):  
GUILLEM PERARNAU ◽  
ORIOL SERRA
Keyword(s):  
Bipartite Graph ◽  
High Probability ◽  
Perfect Matching ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Perfect Matchings ◽  
Asymptotic Enumeration ◽  
Complete Bipartite Graphs ◽  
Edge Colourings ◽  
Square Matrix

A perfect matchingMin an edge-coloured complete bipartite graphKn,nis rainbow if no pair of edges inMhave the same colour. We obtain asymptotic enumeration results for the number of rainbow perfect matchings in terms of the maximum number of occurrences of each colour. We also consider two natural models of random edge-colourings ofKn,nand show that if the number of colours is at leastn, then there is with high probability a rainbow perfect matching. This in particular shows that almost every square matrix of ordernin which every entry appearsntimes has a Latin transversal.

scholarly journals A Note on the Warmth of Random Graphs with Given Expected Degrees

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Statistical Mechanics ◽  
Chromatic Number ◽  
Random Graph ◽  
Lower Bounds ◽  
Random Graphs ◽  
Upper Bound ◽  
Degree Sequence ◽  
Graph Model ◽  
Upper And Lower Bounds ◽  
Random Graph Model

We consider the random graph modelG(w)for a given expected degree sequencew=(w1,w2,…,wn). Warmth, introduced by Brightwell and Winkler in the context of combinatorial statistical mechanics, is a graph parameter related to lower bounds of chromatic number. We present new upper and lower bounds on warmth ofG(w). In particular, the minimum expected degree turns out to be an upper bound of warmth when it tends to infinity and the maximum expected degreem=O(nα)with0<α<1/2.

scholarly journals An Exploration of the Permanent-Determinant Method

10.37236/1384 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Greg Kuperberg
Keyword(s):  
Bipartite Graph ◽  
Perfect Matchings ◽  
New Techniques ◽  
Enumerative Combinatorics ◽  
Plane Graphs ◽  
Quotient Graph ◽  
Plane Partitions ◽  
Symmetry Classes ◽  
Edge Graph

The permanent-determinant method and its generalization, the Hafnian-Pfaffian method, are methods to enumerate perfect matchings of plane graphs that were discovered by P. W. Kasteleyn. We present several new techniques and arguments related to the permanent-determinant with consequences in enumerative combinatorics. Here are some of the results that follow from these techniques: 1. If a bipartite graph on the sphere with $4n$ vertices is invariant under the antipodal map, the number of matchings is the square of the number of matchings of the quotient graph. 2. The number of matchings of the edge graph of a graph with vertices of degree at most 3 is a power of 2. 3. The three Carlitz matrices whose determinants count $a \times b \times c$ plane partitions all have the same cokernel. 4. Two symmetry classes of plane partitions can be enumerated with almost no calculation.

scholarly journals On Star Forest Ascending Subgraph Decomposition

10.37236/5380 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Josep M. Aroca ◽  
Anna Lladó
Keyword(s):  
Bipartite Graph ◽  
Degree Sequence ◽  
Necessary Condition ◽  
Stable Sets ◽  
Ascending Subgraph Decomposition ◽  
Edge Decomposition

The Ascending Subgraph Decomposition (ASD) Conjecture asserts that every graph $G$ with ${n+1\choose 2}$ edges admits an edge decomposition $G=H_1\oplus\cdots \oplus H_n$ such that $H_i$ has $i$ edges and it is isomorphic to a subgraph of $H_{i+1}$, $i=1,\ldots ,n-1$. We show that every bipartite graph $G$ with ${n+1\choose 2}$ edges such that the degree sequence $d_1,\ldots ,d_k$ of one of the stable sets satisfies $ d_{k-i}\ge n-i\; \text{for each}\; 0\le i\le k-1$, admits an ascending subgraph decomposition with star forests. We also give a necessary condition on the degree sequence which is not far from the above sufficient one.

scholarly journals Spectral Extrema for Graphs: The Zarankiewicz Problem

10.37236/212 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
László Babai ◽  
Barry Guiduli
Keyword(s):  
Bipartite Graph ◽  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Complete Bipartite Graph ◽  
Average Degree ◽  
Largest Eigenvalue ◽  
Zarankiewicz Problem ◽  
The Largest Eigenvalue

Let $G$ be a graph on $n$ vertices with spectral radius $\lambda$ (this is the largest eigenvalue of the adjacency matrix of $G$). We show that if $G$ does not contain the complete bipartite graph $K_{t ,s}$ as a subgraph, where $2\le t \le s$, then $$\lambda \le \Big((s-1)^{1/t }+o(1)\Big)n^{1-1/t }$$ for fixed $t$ and $s$ while $n\to\infty$. Asymptotically, this bound matches the Kővári-Turán-Sós upper bound on the average degree of $G$ (the Zarankiewicz problem).

scholarly journals Certain Bounds of Regularity of Elimination Ideals on Operations of Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Zongming Lv ◽  
Muhammad Junaid Ali Junjua ◽  
Muhammad Tajammal Tahir ◽  
Khurram Shabbir
Keyword(s):  
Upper Bound ◽  
Degree Sequence ◽  
Borel Type

Elimination ideals are regarded as a special type of Borel type ideals, obtained from degree sequence of a graph, introduced by Anwar and Khalid. In this paper, we compute graphical degree stabilities of K n ∨ C m and K n ∗ C m by using the DVE method. We further compute sharp upper bound for Castelnuovo–Mumford regularity of elimination ideals associated to these families of graphs.

Graph Degree Sequence Solely Determines the Expected Hopfield Network Pattern Stability

2015 ◽  
Vol 27 (1) ◽  
pp. 202-210 ◽  
Author(s):  
Daniel Berend ◽  
Shlomi Dolev ◽  
Ariel Hanemann
Keyword(s):  
Degree Sequence ◽  
Hopfield Neural Network ◽  
Hopfield Network ◽  
Standard Normal Distribution ◽  
Normal Distribution Function ◽  
Instability Point ◽  
Pattern Stability ◽  
Standard Normal ◽  
Network Pattern ◽  
General Graphs

We analyze the effect of network topology on the pattern stability of the Hopfield neural network in the case of general graphs. The patterns are randomly selected from a uniform distribution. We start the Hopfield procedure from some pattern v. An error in an entry e of v is the situation where, if the procedure is started at e, the value of e flips. Such an entry is an instability point. Note that we disregard the value at e by the end of the procedure, as well as what happens if we start the procedure from another pattern [Formula: see text] or another entry [Formula: see text] of v. We measure the instability of the system by the expected total number of instability points of all the patterns. Our main result is that the instability of the system does not depend on the exact topology of the underlying graph, but rather only on its degree sequence. Moreover, for a large number of nodes, the instability can be approximated by [Formula: see text], where [Formula: see text] is the standard normal distribution function and [Formula: see text] are the degrees of the nodes.

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