scholarly journals Rainbow Matchings and Rainbow Connectedness

10.37236/5246 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Alexey Pokrovskiy
Keyword(s):  
Bipartite Graph ◽  
Directed Graphs ◽  
Latin Squares ◽  
Independent Interest ◽  
Edge Disjoint ◽  
Rainbow Matching

Aharoni and Berger conjectured that every collection of $n$ matchings of size $n+1$ in a bipartite graph contains a rainbow matching of size $n$. This conjecture is related to several old conjectures of Ryser, Brualdi, and Stein about transversals in Latin squares. There have been many recent partial results about the Aharoni-Berger Conjecture. The conjecture is known to hold when the matchings are much larger than $n+1$. The best bound is currently due to Aharoni, Kotlar, and Ziv who proved the conjecture when the matchings are of size at least $3n/2+1$. When the matchings are all edge-disjoint and perfect, the best result follows from a theorem of Häggkvist and Johansson which implies the conjecture when the matchings have size at least $n+o(n)$. In this paper we show that the conjecture is true when the matchings have size $n+o(n)$ and are all edge-disjoint (but not necessarily perfect). We also give an alternative argument to prove the conjecture when the matchings have size at least $\phi n+o(n)$ where $\phi\approx 1.618$ is the Golden Ratio.Our proofs involve studying connectedness in coloured, directed graphs. The notion of connectedness that we introduce is new, and perhaps of independent interest.

scholarly journals An approximate version of Jackson’s conjecture

2020 ◽  
Vol 29 (6) ◽  
pp. 886-899
Author(s):  
Anita Liebenau ◽  
Yanitsa Pehova
Keyword(s):  
Bipartite Graph ◽  
Small Proportion ◽  
Complete Bipartite Graph ◽  
Hamilton Cycle ◽  
Hamilton Cycles ◽  
Edge Disjoint ◽  
Bipartite Digraph ◽  
Bipartite Tournament ◽  
Approximate Version

AbstractA diregular bipartite tournament is a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and out-degree. In 1981 Jackson showed that a diregular bipartite tournament contains a Hamilton cycle, and conjectured that in fact its edge set can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: for every ε > 0 there exists n0 such that every diregular bipartite tournament on 2n ≥ n0 vertices contains a collection of (1/2–ε)n cycles of length at least (2–ε)n. Increasing the degree by a small proportion allows us to prove the existence of many Hamilton cycles: for every c > 1/2 and ε > 0 there exists n0 such that every cn-regular bipartite digraph on 2n ≥ n0 vertices contains (1−ε)cn edge-disjoint Hamilton cycles.

scholarly journals Perfect Factorisations of Bipartite Graphs and Latin Squares Without Proper Subrectangles

10.37236/1441 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
I. M. Wanless
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Latin Square ◽  
Bipartite Graphs ◽  
Latin Squares ◽  
Existence Results ◽  
Strong Sense ◽  
Single Cycle

A Latin square is pan-Hamiltonian if every pair of rows forms a single cycle. Such squares are related to perfect 1-factorisations of the complete bipartite graph. A square is atomic if every conjugate is pan-Hamiltonian. These squares are indivisible in a strong sense – they have no proper subrectangles. We give some existence results and a catalogue for small orders. In the process we identify all the perfect 1-factorisations of $K_{n,n}$ for $n\leq 9$, and count the Latin squares of order $9$ without proper subsquares.

Edge-disjoint cycles in regular directed graphs

1996 ◽  
Vol 22 (3) ◽  
pp. 231-237 ◽  
Author(s):  
Noga Alon ◽  
Colin McDiarmid ◽  
Michael Molloy
Keyword(s):  
Directed Graphs ◽  
Edge Disjoint ◽  
Disjoint Cycles

scholarly journals Decomposition Spaces and Restriction Species

2018 ◽  
Vol 2020 (21) ◽  
pp. 7558-7616 ◽  
Author(s):  
Imma Gálvez-Carrillo ◽  
Joachim Kock ◽  
Andrew Tonks
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Directed Graphs ◽  
Rooted Trees ◽  
Independent Interest ◽  
General Construction ◽  
Finite Sets ◽  
Decomposition Spaces ◽  
Segal Spaces ◽  
Finite Posets

Abstract We show that Schmitt’s restriction species (such as graphs, matroids, posets, etc.) naturally induce decomposition spaces (a.k.a. unital $2$-Segal spaces), and that their associated coalgebras are an instance of the general construction of incidence coalgebras of decomposition spaces. We introduce directed restriction species that subsume Schmitt’s restriction species and also induce decomposition spaces. Whereas ordinary restriction species are presheaves on the category of finite sets and injections, directed restriction species are presheaves on the category of finite posets and convex maps. We also introduce the notion of monoidal (directed) restriction species, which induce monoidal decomposition spaces and hence bialgebras, most often Hopf algebras. Examples of this notion include rooted forests, directed graphs, posets, double posets, and many related structures. A prominent instance of a resulting incidence bialgebra is the Butcher–Connes–Kreimer Hopf algebra of rooted trees. Both ordinary and directed restriction species are shown to be examples of a construction of decomposition spaces from certain cocartesian fibrations over the category of finite ordinals that are also cartesian over convex maps. The proofs rely on some beautiful simplicial combinatorics, where the notion of convexity plays a key role. The methods developed are of independent interest as techniques for constructing decomposition spaces.

The Coarseness of the Complete Bipartite Graph

1969 ◽  
Vol 21 ◽  
pp. 1086-1096 ◽  
Author(s):  
Lowell W. Beineke ◽  
Richard K. Guy
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Bipartite Graph ◽  
The Other ◽  
Integer Part ◽  
Edge Disjoint ◽  
Image Position

The coarseness, c(G), of a graph G is the maximum number of edge-disjoint, non-planar graphs whose union is G. The coarseness of the complete graph has been investigated elsewhere (1; 2). We consider the coarseness of the complete bipartite, or 2-coloured, graph, Km,n, consisting of sets of mand nvertices, each member of one set being joined by an edge to each member of the other. No members of one set are joined to each other.Our results are summarized in the following theorem, where square brackets denote “integer part”.THEOREM. If m= 3p + d, 0 ≦ d≦ 2, and n = 3q + e, 0 ≦ e ≦ 2, then for d = 0 or 1 and e = 0 or 1,1

scholarly journals Edit Distance and its Computation

10.37236/744 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
József Balogh ◽  
Ryan Martin
Keyword(s):  
Bipartite Graph ◽  
Edit Distance ◽  
Complete Bipartite Graph ◽  
New Method ◽  
Independent Interest ◽  
Regularity Lemma ◽  
Hereditary Property ◽  
Vertex Set ◽  
Hereditary Properties ◽  
Szemeredi's Regularity Lemma

In this paper, we provide a method for determining the asymptotic value of the maximum edit distance from a given hereditary property. This method permits the edit distance to be computed without using Szemerédi's Regularity Lemma directly. Using this new method, we are able to compute the edit distance from hereditary properties for which it was previously unknown. For some graphs $H$, the edit distance from ${\rm Forb}(H)$ is computed, where ${\rm Forb}(H)$ is the class of graphs which contain no induced copy of graph $H$. Those graphs for which we determine the edit distance asymptotically are $H=K_a+E_b$, an $a$-clique with $b$ isolated vertices, and $H=K_{3,3}$, a complete bipartite graph. We also provide a graph, the first such construction, for which the edit distance cannot be determined just by considering partitions of the vertex set into cliques and cocliques. In the process, we develop weighted generalizations of Turán's theorem, which may be of independent interest.

Graceful Labeling for Some Complete Bipartite Graph

2018 ◽  
Vol 9 (12) ◽  
pp. 2147-2152
Author(s):  
V. Raju ◽  
M. Paruvatha vathana
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Graceful Labeling

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

2020 ◽  
Vol 17 (2) ◽  
pp. 256-277
Author(s):  
Ol'ga Veselovska ◽  
Veronika Dostoina
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Analytic Functions ◽  
Combinatorial Identities ◽  
Independent Interest ◽  
Closed Curves ◽  
In Series ◽  
Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

On the Crossing Number of $K_{m,n}$

10.37236/1748 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Nagi H. Nahas
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.

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