A NOTE ON ALMOST BALANCED BIPARTITIONS OF A GRAPH

2014 ◽  
Vol 91 (2) ◽  
pp. 177-182
Author(s):  
XIAOLAN HU ◽  
YUNQING ZHANG ◽  
YAOJUN CHEN
Keyword(s):  
Bipartite Graph ◽  
Minimum Degree ◽  
Complete Bipartite Graph ◽  
Discrete Math ◽  
Graph Partitions ◽  
Delta G ◽  
Minimum Degrees

AbstractLet $G$ be a graph of order $n\geq 6$ with minimum degree ${\it\delta}(G)\geq 4$. Arkin and Hassin [‘Graph partitions with minimum degree constraints’, Discrete Math. 190 (1998), 55–65] conjectured that there exists a bipartition $S,T$ of $V(G)$ such that $\lfloor n/2\rfloor -2\leq |S|,|T|\leq \lceil n/2\rceil +2$ and the minimum degrees in the subgraphs induced by $S$ and $T$ are at least two. In this paper, we first show that $G$ has a bipartition such that the minimum degree in each part is at least two, and then prove that the conjecture is true if the complement of $G$ contains no complete bipartite graph $K_{3,r}$, where $r=\lfloor n/2\rfloor -3$.

scholarly journals Degree Ramsey Numbers of Closed Blowups of Trees

10.37236/2526 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Paul Horn ◽  
Kevin G. Milans ◽  
Vojtěch Rödl
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Edge Coloring ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Delta G ◽  
Monochromatic Copy

The degree Ramsey number of a graph $G$, denoted $R_\Delta(G;s)$, is $\min\{\Delta(H)\colon\, H\stackrel{s}{\to} G\}$, where $H\stackrel{s}{\to} G$ means that every $s$-edge-coloring of $H$ contains a monochromatic copy of $G$.  The closed $k$-blowup of a graph is obtained by replacing every vertex with a clique of size $k$ and every edge with a complete bipartite graph where both partite sets have size $k$.  We prove that there is a function $f$ such that $R_\Delta(G;s) \le f(\Delta(G), s)$  when $G$ is a closed blowup of a tree.

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 Maximizing the Number of Independent Sets of a Fixed Size

2014 ◽  
Vol 24 (3) ◽  
pp. 521-527 ◽  
Author(s):  
WENYING GAN ◽  
PO-SHEN LOH ◽  
BENNY SUDAKOV
Keyword(s):  
Bipartite Graph ◽  
Minimum Degree ◽  
Complete Bipartite Graph ◽  
Short Note ◽  
Independent Sets ◽  
Fixed Size

Letit(G) be the number of independent sets of sizetin a graphG. Engbers and Galvin asked how largeit(G) could be in graphs with minimum degree at least δ. They further conjectured that whenn⩾ 2δ andt⩾ 3,it(G) is maximized by the complete bipartite graphKδ,n−δ. This conjecture has recently drawn the attention of many researchers. In this short note, we prove this conjecture.

scholarly journals Note on Path-Connectivity of Complete Bipartite Graphs

2021 ◽  
pp. 2142014
Author(s):  
Xiaoxue Gao ◽  
Shasha Li ◽  
Yan Zhao
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Discrete Math ◽  
Bipartite Graphs ◽  
Complete Bipartite Graphs

For a graph [Formula: see text] and a set [Formula: see text] of size at least [Formula: see text], a path in [Formula: see text] is said to be an [Formula: see text]-path if it connects all vertices of [Formula: see text]. Two [Formula: see text]-paths [Formula: see text] and [Formula: see text] are said to be internally disjoint if [Formula: see text] and [Formula: see text]. Let [Formula: see text] denote the maximum number of internally disjoint [Formula: see text]-paths in [Formula: see text]. The [Formula: see text]-path-connectivity [Formula: see text] of [Formula: see text] is then defined as the minimum [Formula: see text], where [Formula: see text] ranges over all [Formula: see text]-subsets of [Formula: see text]. In [M. Hager, Path-connectivity in graphs, Discrete Math. 59 (1986) 53–59], the [Formula: see text]-path-connectivity of the complete bipartite graph [Formula: see text] was calculated, where [Formula: see text]. But, from his proof, only the case that [Formula: see text] was considered. In this paper, we calculate the situation that [Formula: see text] and complete the result.

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.

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 .

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