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 Edge Disjoint Hamilton Cycles in Knödel Graphs

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Palanivel Subramania Nadar Paulraja ◽  
S Sampath Kumar
Keyword(s):  
Maximum Degree ◽  
Hamilton Cycle ◽  
Hamilton Cycles ◽  
Cycle Decomposition ◽  
Edge Disjoint ◽  
International Audience ◽  
Appl Math

International audience The vertices of the Knödel graph $W_{\Delta, n}$ on $n \geq 2$ vertices, $n$ even, and of maximum degree $\Delta, 1 \leq \Delta \leq \lfloor log_2(n) \rfloor$, are the pairs $(i,j)$ with $i=1,2$ and $0 \leq j \leq \frac{n}{2} -1$. For $0 \leq j \leq \frac{n}{2} -1$, there is an edge between vertex $(1,j)$ and every vertex $(2,j + 2^k - 1 (mod \frac{n}{2}))$, for $k=0,1,2, \ldots , \Delta -1$. Existence of a Hamilton cycle decomposition of $W_{k, 2k}, k \geq 6$ is not yet known, see Discrete Appl. Math. 137 (2004) 173-195. In this paper, it is shown that the $k$-regular Knödel graph $W_{k,2k}, k \geq 6$ has $ \lfloor \frac{k}{2} \rfloor - 1$ edge disjoint Hamilton cycles.

scholarly journals Matchings and Hamilton cycles in hypergraphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Daniela Kühn ◽  
Deryk Osthus
Keyword(s):  
Bipartite Graph ◽  
Perfect Matching ◽  
Minimum Degree ◽  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Uniform Hypergraphs ◽  
International Audience

International audience It is well known that every bipartite graph with vertex classes of size $n$ whose minimum degree is at least $n/2$ contains a perfect matching. We prove an analogue of this result for uniform hypergraphs. We also provide an analogue of Dirac's theorem on Hamilton cycles for $3$-uniform hypergraphs: We say that a $3$-uniform hypergraph has a Hamilton cycle if there is a cyclic ordering of its vertices such that every pair of consecutive vertices lies in a hyperedge which consists of three consecutive vertices. We prove that for every $\varepsilon > 0$ there is an $n_0$ such that every $3$-uniform hypergraph of order $n \geq n_0$ whose minimum degree is at least $n/4+ \varepsilon n$ contains a Hamilton cycle. Our bounds on the minimum degree are essentially best possible.

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 Packing Loose Hamilton Cycles

2017 ◽  
Vol 26 (6) ◽  
pp. 839-849
Author(s):  
ASAF FERBER ◽  
KYLE LUH ◽  
DANIEL MONTEALEGRE ◽  
OANH NGUYEN
Keyword(s):  
Packing Problem ◽  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Threshold Probability ◽  
Edge Disjoint ◽  
Formula Group ◽  
Vertex Set ◽  
Substantial Progress ◽  
Image Position

A subsetCof edges in ak-uniform hypergraphHis aloose Hamilton cycleifCcovers all the vertices ofHand there exists a cyclic ordering of these vertices such that the edges inCare segments of that order and such that every two consecutive edges share exactly one vertex. The binomial randomk-uniform hypergraphHkn,phas vertex set [n] and an edge setEobtained by adding eachk-tuplee∈ ($\binom{[n]}{k}$) toEwith probabilityp, independently at random.Here we consider the problem of finding edge-disjoint loose Hamilton cycles covering all buto(|E|) edges, referred to as thepacking problem. While it is known that the threshold probability of the appearance of a loose Hamilton cycle inHkn,pis$p=\Theta\biggl(\frac{\log n}{n^{k-1}}\biggr),$the best known bounds for the packing problem are aroundp= polylog(n)/n. Here we make substantial progress and prove the following asymptotically (up to a polylog(n) factor) best possible result: forp≥ logCn/nk−1, a randomk-uniform hypergraphHkn,pwith high probability contains$N:=(1-o(1))\frac{\binom{n}{k}p}{n/(k-1)}$edge-disjoint loose Hamilton cycles.Our proof utilizes and modifies the idea of ‘online sprinkling’ recently introduced by Vu and the first author.

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

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 $n$.

scholarly journals Join Products K2,3 + Cn

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 925
Author(s):  
Michal Staš
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number ◽  
Arithmetic Means ◽  
Minimum Number ◽  
Edge Crossings

The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

scholarly journals Counting Hamilton cycles in Dirac hypergraphs

2020 ◽  
pp. 1-23
Author(s):  
Stefan Glock ◽  
Stephen Gould ◽  
Felix Joos ◽  
Daniela Kühn ◽  
Deryk Osthus
Keyword(s):  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Similar Estimate ◽  
Hamilton Cycles

Abstract A tight Hamilton cycle in a k-uniform hypergraph (k-graph) G is a cyclic ordering of the vertices of G such that every set of k consecutive vertices in the ordering forms an edge. Rödl, Ruciński and Szemerédi proved that for $k\ge 3$ , every k-graph on n vertices with minimum codegree at least $n/2+o(n)$ contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such k-graphs is ${\exp(n\ln n-\Theta(n))}$ . As a corollary, we obtain a similar estimate on the number of Hamilton ${\ell}$ -cycles in such k-graphs for all ${\ell\in\{0,\ldots,k-1\}}$ , which makes progress on a question of Ferber, Krivelevich and Sudakov.

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