scholarly journals Decomposing Dense Bipartite Graphs into 4-Cycles

10.37236/3444 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Nicholas Cavenagh
Keyword(s):  
Bipartite Graph ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Degree Of A Vertex

Let $G$ be an even bipartite graph with partite sets $X$ and $Y$ such that $|Y|$ is even and the minimum degree of a vertex in $Y$ is at least $95|X|/96$. Suppose furthermore that the number of edges in $G$ is divisible by $4$. Then $G$ decomposes into 4-cycles.

scholarly journals Detection number of bipartite graphs and cubic graphs

2014 ◽  
Vol Vol. 16 no. 3 ◽  
Author(s):  
Frederic Havet ◽  
Nagarajan Paramaguru ◽  
Rathinaswamy Sampathkumar
Keyword(s):  
Positive Integer ◽  
Bipartite Graph ◽  
Connected Graph ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
International Audience ◽  
Np Complete

International audience For a connected graph G of order |V(G)| ≥3 and a k-labelling c : E(G) →{1,2,…,k} of the edges of G, the code of a vertex v of G is the ordered k-tuple (ℓ1,ℓ2,…,ℓk), where ℓi is the number of edges incident with v that are labelled i. The k-labelling c is detectable if every two adjacent vertices of G have distinct codes. The minimum positive integer k for which G has a detectable k-labelling is the detection number det(G) of G. In this paper, we show that it is NP-complete to decide if the detection number of a cubic graph is 2. We also show that the detection number of every bipartite graph of minimum degree at least 3 is at most 2. Finally, we give some sufficient condition for a cubic graph to have detection number 3.

scholarly journals Some new sufficient conditions for 2p-Hamilton-biconnectedness of graphs

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 993-1011
Author(s):  
Ming-Zhu Chen ◽  
Xiao-Dong Zhang
Keyword(s):  
Bipartite Graph ◽  
Minimum Degree ◽  
Sufficient Conditions ◽  
Bipartite Graphs

A balanced bipartite graph G is said to be 2p-Hamilton-biconnected if for any balanced subset W of size 2p of V(G), the subgraph induced by V(G)nW is Hamilton-biconnected. In this paper, we prove that ?Let G be a balanced bipartite graph of order 2n with minimum degree ?(G) ? k, where n ? 2k-p+2 for two integers k ? p ? 0. If the number of edges e(G) > n(n-k + p-1) + (k + 2)(k-p+1), then G is 2p-Hamilton-biconnected except some exceptions.? Furthermore, this result is used to present two new spectral conditions for a graph to be 2p-Hamilton-biconnected. Moreover, the similar results are also presented for nearly balanced bipartite graphs.

scholarly journals Embedding Spanning Bipartite Graphs of Small Bandwidth

2012 ◽  
Vol 22 (1) ◽  
pp. 71-96 ◽  
Author(s):  
FIACHRA KNOX ◽  
ANDREW TREGLOWN
Keyword(s):  
Bipartite Graph ◽  
Minimum Degree ◽  
Degree Sequence ◽  
Bipartite Graphs ◽  
Type Result ◽  
Bounded Degree ◽  
Expansion Property ◽  
Small Bandwidth ◽  
Bipartite Case

Böttcher, Schacht and Taraz (Math. Ann., 2009) gave a condition on the minimum degree of a graph G on n vertices that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth o(n), thereby proving a conjecture of Bollobás and Komlós (Combin. Probab. Comput., 1999). We strengthen this result in the case when H is bipartite. Indeed, we give an essentially best-possible condition on the degree sequence of a graph G on n vertices that forces G to contain every bipartite graph H on n vertices of bounded degree and of bandwidth o(n). This also implies an Ore-type result. In fact, we prove a much stronger result where the condition on G is relaxed to a certain robust expansion property. Our result also confirms the bipartite case of a conjecture of Balogh, Kostochka and Treglown concerning the degree sequence of a graph which forces a perfect H-packing.

scholarly journals Spanning $k$-trees of Bipartite Graphs

10.37236/3628 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Mikio Kano ◽  
Kenta Ozeki ◽  
Kazuhiro Suzuki ◽  
Masao Tsugaki ◽  
Tomoki Yamashita
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Degree Sum

A tree is called a $k$-tree if its maximum degree is at most $k$. We prove the following theorem. Let $k \geq 2$ be an integer, and $G$ be a connected bipartite graph with bipartition $(A,B)$ such that $|A| \le |B| \le (k-1)|A|+1$. If $\sigma_k(G) \ge |B|$, then $G$ has a spanning $k$-tree, where $\sigma_k(G)$ denotes the minimum degree sum of $k$ independent vertices of $G$. Moreover, the condition on $\sigma_k(G)$ is sharp. It was shown by Win (Abh. Math. Sem. Univ. Hamburg, 43, 263–267, 1975) that if a connected graph $H$ satisfies $\sigma_k(H) \ge |H|-1$, then $H$ has a spanning $k$-tree. Thus our theorem shows that the condition becomes much weaker if the graph is bipartite.

scholarly journals On $k$-Ordered Bipartite Graphs

10.37236/1704 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Jill R. Faudree ◽  
Ronald J. Gould ◽  
Florian Pfender ◽  
Allison Wolf
Keyword(s):  
Positive Integer ◽  
Bipartite Graph ◽  
Hamiltonian Cycle ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Degree Conditions ◽  
The Given

In 1997, Ng and Schultz introduced the idea of cycle orderability. For a positive integer $k$, a graph $G$ is k-ordered if for every ordered sequence of $k$ vertices, there is a cycle that encounters the vertices of the sequence in the given order. If the cycle is also a hamiltonian cycle, then $G$ is said to be k-ordered hamiltonian. We give minimum degree conditions and sum of degree conditions for nonadjacent vertices that imply a balanced bipartite graph to be $k$-ordered hamiltonian. For example, let $G$ be a balanced bipartite graph on $2n$ vertices, $n$ sufficiently large. We show that for any positive integer $k$, if the minimum degree of $G$ is at least $(2n+k-1)/4$, then $G$ is $k$-ordered hamiltonian.

scholarly journals Transversals and Bipancyclicity in Bipartite Graph Families

10.37236/9489 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Peter Bradshaw
Keyword(s):  
Bipartite Graph ◽  
Perfect Matching ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
The Other ◽  
Color Class ◽  
Degree Conditions ◽  
Graph Families ◽  
Balanced Bipartition

A bipartite graph is called bipancyclic if it contains cycles of every even length from four up to the number of vertices in the graph. A theorem of Schmeichel and Mitchem states that for $n \geqslant 4$, every balanced bipartite graph on $2n$ vertices in which each vertex in one color class has degree greater than $\frac{n}{2}$ and each vertex in the other color class has degree at least $\frac{n}{2}$ is bipancyclic. We prove a generalization of this theorem in the setting of graph transversals. Namely, we show that given a family $\mathcal{G}$ of $2n$ bipartite graphs on a common set $X$ of $2n$ vertices with a common balanced bipartition, if each graph of $\mathcal G$ has minimum degree greater than $\frac{n}{2}$ in one color class and minimum degree at least $\frac{n}{2}$ in the other color class, then there exists a cycle on $X$ of each even length $4 \leqslant \ell \leqslant 2n$ that uses at most one edge from each graph of $\mathcal G$. We also show that given a family $\mathcal G$ of $n$ bipartite graphs on a common set $X$ of $2n$ vertices meeting the same degree conditions, there exists a perfect matching on $X$ that uses exactly one edge from each graph of $\mathcal G$.

scholarly journals Regular Spanning Subgraphs of Bipartite Graphs of High Minimum Degree

10.37236/1022 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Béla Csaba
Keyword(s):  
Bipartite Graph ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Spanning Subgraphs ◽  
Spanning Subgraph ◽  
Delta G

Let $G$ be a simple balanced bipartite graph on $2n$ vertices, $\delta = \delta(G)/n$, and $\rho_0={\delta + \sqrt{2 \delta -1} \over 2}$. If $\delta \ge 1/2$ then $G$ has a $\lfloor \rho_0 n \rfloor$-regular spanning subgraph. The statement is nearly tight.

scholarly journals On the induced matching problem in hamiltonian bipartite graphs

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yinglei Song
Keyword(s):  
Bipartite Graph ◽  
Parameterized Complexity ◽  
Bipartite Graphs ◽  
Matching Problem ◽  
Induced Matching ◽  
Subexponential Time ◽  
Maximum Induced Matching

Abstract In this paper, we study the parameterized complexity of the induced matching problem in hamiltonian bipartite graphs and the inapproximability of the maximum induced matching problem in hamiltonian bipartite graphs. We show that, given a hamiltonian bipartite graph, the induced matching problem is W[1]-hard and cannot be solved in time n o ⁢ ( k ) {n^{o(\sqrt{k})}} , where n is the number of vertices in the graph, unless the 3SAT problem can be solved in subexponential time. In addition, we show that unless NP = P {\operatorname{NP}=\operatorname{P}} , a maximum induced matching in a hamiltonian bipartite graph cannot be approximated within a ratio of n 1 / 4 - ϵ {n^{1/4-\epsilon}} , where n is the number of vertices in the graph.

scholarly journals On the Relation of Separability, Bandwidth and Embedding

2019 ◽  
Vol 35 (6) ◽  
pp. 1541-1553
Author(s):  
Béla Csaba ◽  
Bálint Vásárhelyi
Keyword(s):  
Minimum Degree ◽  
Bipartite Graphs ◽  
Bounded Degree ◽  
Spanning Subgraphs ◽  
Large Bandwidth

Abstract In this paper we construct a class of bounded degree bipartite graphs with a small separator and large bandwidth, thereby showing that separability and bandwidth are not linearly equivalent. Furthermore, we also prove that graphs from this class are spanning subgraphs of graphs with minimum degree just slightly above n / 2,  even though their bandwidth is large.

Spectral conditions and Hamiltonicity of a balanced bipartite graph with large minimum degree

2019 ◽  
Vol 356 ◽  
pp. 137-143
Author(s):  
Gui-Sheng Jiang ◽  
Gui-Dong Yu ◽  
Yi Fang
Keyword(s):  
Bipartite Graph ◽  
Minimum Degree
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