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 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 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 Bipartite graphs with close domination and k-domination numbers

2020 ◽  
Vol 18 (1) ◽  
pp. 873-885
Author(s):  
Gülnaz Boruzanlı Ekinci ◽  
Csilla Bujtás
Keyword(s):  
Positive Integer ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{Δ}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

scholarly journals Semi-Degree Threshold for Anti-Directed Hamiltonian Cycles

10.37236/3610 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Louis DeBiasio ◽  
Theodore Molla
Keyword(s):  
Directed Graph ◽  
Hamiltonian Cycle ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Hamiltonian Cycles ◽  
Alternate Direction ◽  
Degree Conditions

In 1960 Ghouila-Houri extended Dirac's theorem to directed graphs by proving that if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $n/2$, then $D$ contains a directed Hamiltonian cycle. For directed graphs one may ask for other orientations of a Hamiltonian cycle and in 1980 Grant initiated the problem of determining minimum degree conditions for a directed graph $D$ to contain an anti-directed Hamiltonian cycle (an orientation in which consecutive edges alternate direction). We prove that for sufficiently large even $n$, if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $\frac{n}{2}+1$, then $D$ contains an anti-directed Hamiltonian cycle. In fact, we prove the stronger result that $\frac{n}{2}$ is sufficient unless $D$ is one of two counterexamples. This result is sharp.

Cover Product and Betti Polynomial of Graphs

2015 ◽  
Vol 58 (2) ◽  
pp. 320-333
Author(s):  
Aurora Llamas ◽  
Josá Martínez–Bernal
Keyword(s):  
Positive Integer ◽  
Bipartite Graph ◽  
Betti Numbers ◽  
Recursive Formula ◽  
Bipartite Graphs ◽  
Matching Number ◽  
Edge Ideal ◽  
Graded Betti Numbers ◽  
Vertex Set ◽  
Chordal Bipartite Graphs

AbstractThe cover product of disjoint graphs G and H with fixed vertex covers C(G) and C(H), is the graphwith vertex set V(G) ∪ V(H) and edge setWe describe the graded Betti numbers of GeH in terms of those of. As applications we obtain: (i) For any positive integer k there exists a connected bipartite graph G such that reg R/I(G) = μS(G) + k, where, I(G) denotes the edge ideal of G, reg R/I(G) is the Castelnuovo–Mumford regularity of R/I(G) and μS(G) is the induced or strong matching number of G; (ii)The graded Betti numbers of the complement of a tree depends only upon its number of vertices; (iii)The h-vector of R/I(G e H) is described in terms of the h-vectors of R/I(G) and R/I(H). Furthermore, in a diòerent direction, we give a recursive formula for the graded Betti numbers of chordal bipartite graphs.

Ramsey Problems with Bounded Degree Spread

1993 ◽  
Vol 2 (3) ◽  
pp. 263-269 ◽  
Author(s):  
G. Chen ◽  
R. H. Schelp
Keyword(s):  
Positive Integer ◽  
Complete Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Bounded Degree ◽  
Monochromatic Copy

Let k be a positive integer, k ≥ 2. In this paper we study bipartite graphs G such that, for n sufficiently large, each two-coloring of the edges of the complete graph Kn gives a monochromatic copy of G, with some k of its vertices having the maximum degree of these k vertices minus the minimum degree of these k vertices (in the colored Kn) at most k − 2.

AN ALGORITHM FOR FINDING LONGEST CYCLES IN CERTAIN BIPARTITE GRAPHS

2001 ◽  
Vol 12 (04) ◽  
pp. 445-454
Author(s):  
PAK-KEN WONG
Keyword(s):  
Bipartite Graph ◽  
Efficient Algorithm ◽  
Hamiltonian Cycle ◽  
Bipartite Graphs ◽  
Longest Cycles

Let G be a connected bipartite graph with with bipartition (X, Y) such that |X| ≥ |Y| (≥ 2). Put n = |X|, m = |Y| and l = m + n. Suppose that, for all vertices x ∈ X and y ∈ Y, dist(x,y) = 3 implies d(x) + d(y) ≥ n + 1. We show that G contains a cycle of length 2m. We also give an efficient algorithm to obtain such a cycle. The complexity of this algorithm is O(l3). In case m = n, we find a hamiltonian cycle of G. This generalizes a result given in [10].

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.

The Genus of a Random Bipartite Graph

2019 ◽  
Vol 72 (6) ◽  
pp. 1607-1623 ◽  
Author(s):  
Yifan Jing ◽  
Bojan Mohar
Keyword(s):  
Phase Transitions ◽  
Positive Integer ◽  
Bipartite Graph ◽  
Random Graph ◽  
Analogous Result ◽  
Bipartite Graphs ◽  
Constant Size

AbstractArchdeacon and Grable (1995) proved that the genus of the random graph $G\in {\mathcal{G}}_{n,p}$ is almost surely close to $pn^{2}/12$ if $p=p(n)\geqslant 3(\ln n)^{2}n^{-1/2}$. In this paper we prove an analogous result for random bipartite graphs in ${\mathcal{G}}_{n_{1},n_{2},p}$. If $n_{1}\geqslant n_{2}\gg 1$, phase transitions occur for every positive integer $i$ when $p=\unicode[STIX]{x1D6E9}((n_{1}n_{2})^{-i/(2i+1)})$. A different behaviour is exhibited when one of the bipartite parts has constant size, i.e., $n_{1}\gg 1$ and $n_{2}$ is a constant. In that case, phase transitions occur when $p=\unicode[STIX]{x1D6E9}(n_{1}^{-1/2})$ and when $p=\unicode[STIX]{x1D6E9}(n_{1}^{-1/3})$.

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