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 Komlós’ tiling theorem in random graphs

2019 ◽  
Vol 29 (1) ◽  
pp. 113-127
Author(s):  
Rajko Nenadov ◽  
Nemanja Škorić
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Random Graphs ◽  
Upper Bound ◽  
Minimum Degree ◽  
Arbitrary Graph ◽  
Spanning Subgraph ◽  
Delta G ◽  
Image Position ◽  
Vertex Disjoint

AbstractGiven graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if $p \ge C{n^{ - 1/{m_2}(H)}}$ , then asymptotically almost surely every spanning subgraph G of the random graph 𝒢(n, p) with minimum degree at least $\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}}(H)}} + \gamma )np$ contains an H-tiling that covers all but at most γn vertices. Here, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlós, and m2(H) is the 2-density of H. We show that this theorem can be bootstrapped to obtain an H-tiling covering all but at most $\gamma {(C/p)^{{m_2}(H)}}$ vertices, which is strictly smaller when $p \ge C{n^{ - 1/{m_2}(H)}}$ . In the case where H = K3, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph H we give an upper bound on p for which some leftover is unavoidable and a bound on the size of a largest H -tiling for p below this value.

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.

scholarly journals A Note on Path Factors of $(3,4)$-Biregular Bipartite Graphs

10.37236/705 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Carl Johan Casselgren
Keyword(s):  
Graph Theory ◽  
Bipartite Graph ◽  
Edge Coloring ◽  
Bipartite Graphs ◽  
The Other ◽  
Spanning Subgraph ◽  
Interval Coloring ◽  
Proper Edge Coloring

A proper edge coloring of a graph $G$ with colors $1,2,3,\dots$ is called an interval coloring if the colors on the edges incident with any vertex are consecutive. A bipartite graph is $(3,4)$-biregular if all vertices in one part have degree $3$ and all vertices in the other part have degree $4$. Recently it was proved [J. Graph Theory 61 (2009), 88-97] that if such a graph $G$ has a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in $\{2, 4, 6, 8\}$, then $G$ has an interval coloring. It was also conjectured that every simple $(3,4)$-biregular bipartite graph has such a subgraph. We provide some evidence for this conjecture by proving that a simple $(3,4)$-biregular bipartite graph has a spanning subgraph whose components are nontrivial paths with endpoints at $3$-valent vertices and lengths not exceeding $22$.

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 Independent Sets in Graphs with Given Minimum Degree

10.37236/2722 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
James Alexander ◽  
Jonathan Cutler ◽  
Tim Mink
Keyword(s):  
Bipartite Graph ◽  
Minimum Degree ◽  
Independent Sets ◽  
Regular Graphs ◽  
Double Covers ◽  
Delta G ◽  
Regular Bipartite Graph

The enumeration of independent sets in graphs with various restrictions has been a topic of much interest of late.  Let $i(G)$ be the number of independent sets in a graph $G$ and let $i_t(G)$ be the number of independent sets in $G$ of size $t$.  Kahn used entropy to show that if $G$ is an $r$-regular bipartite graph with $n$ vertices, then $i(G)\leq i(K_{r,r})^{n/2r}$.  Zhao used bipartite double covers to extend this bound to general $r$-regular graphs.  Galvin proved that if $G$ is a graph with $\delta(G)\geq \delta$ and $n$ large enough, then $i(G)\leq i(K_{\delta,n-\delta})$.  In this paper, we prove that if $G$ is a bipartite graph on $n$ vertices with $\delta(G)\geq\delta$ where $n\geq 2\delta$, then $i_t(G)\leq i_t(K_{\delta,n-\delta})$ when $t\geq 3$.  We note that this result cannot be extended to $t=2$ (and is trivial for $t=0,1$).  Also, we use Kahn's entropy argument and Zhao's extension to prove that if $G$ is a graph with $n$ vertices, $\delta(G)\geq\delta$, and $\Delta(G)\leq \Delta$, then $i(G)\leq i(K_{\delta,\Delta})^{n/2\delta}$.

scholarly journals On a Class of Polynomials Associated with the Stars of a Graph and its Application to Node-Disjoint Decompositions of Complete Graphs and Complete Bipartite Graphs into Stars

1979 ◽  
Vol 22 (1) ◽  
pp. 35-46 ◽  
Author(s):  
E. J. Farrell
Keyword(s):  
Connected Graph ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Spanning Subgraphs ◽  
Spanning Subgraph ◽  
Complete Bipartite Graphs

AbstractA star is a connected graph in which every node but possibly one has valency 1. Let G be a graph and C a spanning subgraph of G in which every component is a star. With each component α of C let us associate a weight wα. Let Пα wα be the weight associated with the entire subgraph G the star polynomial of G is ΣПα wα where the summation is taken over all spanning subgraphs of G consisting of stars. In this paper an algorithm for finding star polynomials of graphs is given. The star polynomials of various classes of graphs are then found, and some results about node-disjoint decomposition of complete graphs and complete bipartite graphs are deduced.

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 $\mathcal G$.

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