On a Conjecture on Directed Cycles in a Directed Bipartite Graph

1997 ◽  
Vol 13 (3) ◽  
pp. 267-273 ◽  
Author(s):  
Charles Little ◽  
Kee Teo ◽  
Hong Wang
Keyword(s):  
Bipartite Graph ◽  
Directed Cycles

scholarly journals A scheduling problem with a simple graphical solution

1980 ◽  
Vol 21 (4) ◽  
pp. 486-495 ◽  
Author(s):  
A. F. Beecham ◽  
A. C. Hurley
Keyword(s):  
Positive Integer ◽  
Bipartite Graph ◽  
Practical Case ◽  
Scheduling Problem ◽  
Graphical Solution ◽  
Marriage Problem ◽  
Practical Applications ◽  
Golf Clubs ◽  
Directed Cycles

AbstractIt is shown that a problem which arose in the scheduling of two simultaneous competitions between a number of golf clubs may be reduced to that of 4- colouring the edges of a certain bipartite graph which has 4 edges meeting at each vertex. This colouring problem is solved by an analysis in terms of directed cycles, which is simple to carry through in a practical case and is easily extended to the problem with 4 replaced by 2m. The more general colouring problem with 4 replaced by any positive integer is solved by relating it to the marriage problem enunciated by Philip Hall and to the latin multiplication technique of Kaufmann but, in practical applications, this approach involves severe computational difficulties.

scholarly journals Partition of a directed bipartite graph into two directed cycles

1996 ◽  
Vol 160 (1-3) ◽  
pp. 283-289 ◽  
Author(s):  
Hong Wang ◽  
Charles Little ◽  
Kee Teo
Keyword(s):  
Bipartite Graph ◽  
Directed Cycles

scholarly journals On the Caccetta-Häggkvist Conjecture with a Forbidden Transitive Tournament

10.37236/5954 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Andrzej Grzesik
Keyword(s):  
Bipartite Graph ◽  
Blow Up ◽  
Oriented Graph ◽  
Extremal Graph ◽  
Directed Cycle ◽  
Extremal Graphs ◽  
Flag Algebras ◽  
Transitive Tournament ◽  
Regular Bipartite Graph ◽  
Directed Cycles

The Caccetta-Häggkvist Conjecture asserts that every oriented graph on $n$ vertices without directed cycles of length less than or equal to $l$ has minimum outdegree at most $(n-1)/l$. In this paper we state a conjecture for graphs missing a transitive tournament on $2^k+1$ vertices, with a weaker assumption on minimum outdegree. We prove that the Caccetta-Häggkvist Conjecture follows from the presented conjecture and show matching constructions for all $k$ and $l$. The main advantage of considering this generalized conjecture is that it reduces the set of the extremal graphs and allows using an induction.We also prove the triangle case of the conjecture for $k=1$ and $2$ by using the Razborov's flag algebras. In particular, it proves the most interesting and studied case of the Caccetta-Häggkvist Conjecture in the class of graphs without the transitive tournament on 5 vertices. It is also shown that the extremal graph for the case $k=2$ has to be a blow-up of a directed cycle on 4 vertices having in each blob an extremal graph for the case $k=1$ (complete regular bipartite graph), which confirms the conjectured structure of the extremal examples.

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 GAGrank: Software for glycosaminoglycan sequence ranking using a bipartite graph model

2021 ◽  
pp. 100093
Author(s):  
John D. Hogan ◽  
Jiandong Wu ◽  
Joshua A. Klein ◽  
Cheng Lin ◽  
Luis Carvalho ◽  
...  
Keyword(s):  
Bipartite Graph ◽  
Graph Model
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