scholarly journals Chip Games and Paintability

10.37236/5723 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Lech Duraj ◽  
Grzegorz Gutowski ◽  
Jakub Kozik
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Uniform Hypergraph ◽  
Natural Family ◽  
Uniform Hypergraphs ◽  
Classical Correspondence ◽  
On Line ◽  
Choice Number ◽  
The Difference

We prove that the paint number of the complete bipartite graph $K_{N,N}$ is $\log N + O(1)$. As a consequence, we get that the difference between the paint number and the choice number of $K_{N,N}$ is $\Theta(\log \log N)$. This answers in the negative the question of Zhu (2009) whether this difference, for all graphs, can be bounded by a common constant. By a classical correspondence, our result translates to the framework of on-line coloring of uniform hypergraphs. This way we obtain that for every on-line two coloring algorithm there exists a $k$-uniform hypergraph with $\Theta(2^k)$ edges on which the strategy fails. The results are derived through an analysis of a natural family of chip games.

scholarly journals On-Line List Colouring of Graphs

10.37236/216 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Xuding Zhu
Keyword(s):  
Bipartite Graph ◽  
Polynomial Time ◽  
Complete Bipartite Graph ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Line List ◽  
Open Questions ◽  
On Line ◽  
Choice Number

This paper studies on-line list colouring of graphs. It is proved that the on-line choice number of a graph $G$ on $n$ vertices is at most $\chi(G) \ln n+1$, and the on-line $b$-choice number of $G$ is at most ${e\chi(G)-1\over e-1} (b-1+ \ln n)+b$. Suppose $G$ is a graph with a given $\chi(G)$-colouring of $G$. Then for any $(\chi(G) \ln n +1)$-assignment $L$ of $G$, we give a polynomial time algorithm which constructs an $L$-colouring of $G$. For any $({e\chi(G)-1\over e-1} (b-1+ \ln n)+b)$-assignment $L$ of $G$, we give a polynomial time algorithm which constructs an $(L,b)$-colouring of $G$. We then characterize all on-line $2$-choosable graphs. It is also proved that a complete bipartite graph of the form $K_{3,q}$ is on-line $3$-choosable if and only if it is $3$-choosable, but there are graphs of the form $K_{6,q}$ which are $3$-choosable but not on-line $3$-choosable. Some open questions concerning on-line list colouring are posed in the last section.

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.

scholarly journals On a Theorem of Erdős, Rubin, and Taylor on Choosability of Complete Bipartite Graphs

10.37236/1670 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Alexandr Kostochka
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Uniform Hypergraph ◽  
Ordinary Sense ◽  
Minimum Order ◽  
Minimum Number ◽  
Complete Bipartite Graphs

Erdős, Rubin, and Taylor found a nice correspondence between the minimum order of a complete bipartite graph that is not $r$-choosable and the minimum number of edges in an $r$-uniform hypergraph that is not $2$-colorable (in the ordinary sense). In this note we use their ideas to derive similar correspondences for complete $k$-partite graphs and complete $k$-uniform $k$-partite hypergraphs.

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 .

Decomposing Complete 3-Uniform Hypergraph into 5-Cycles

2014 ◽  
Vol 672-674 ◽  
pp. 1935-1939
Author(s):  
Guan Ru Li ◽  
Yi Ming Lei ◽  
Jirimutu
Keyword(s):  
Positive Integer ◽  
Hamiltonian Cycles ◽  
Uniform Hypergraph ◽  
Design Algorithm ◽  
Uniform Hypergraphs ◽  
Edge Partition ◽  
Definition Of

About the Katona-Kierstead definition of a Hamiltonian cycles in a uniform hypergraph, a decomposition of complete k-uniform hypergraph Kn(k) into Hamiltonian cycles studied by Bailey-Stevens and Meszka-Rosa. For n≡2,4,5 (mod 6), we design algorithm for decomposing the complete 3-uniform hypergraphs into Hamiltonian cycles by using the method of edge-partition. A decomposition of Kn(3) into 5-cycles has been presented for all admissible n≤17, and for all n=4m +1, m is a positive integer. In general, the existence of a decomposition into 5-cycles remains open. In this paper, we use the method of edge-partition and cycle sequence proposed by Jirimutu and Wang. We find a decomposition of K20(3) into 5-cycles.

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