The Order of Monochromatic Subgraphs with a Given Minimum Degree

10.37236/1725 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Minimum Degree

Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order at least $t$. Let $f_G(d)=0$ in case there is a $2$-coloring of the edges of $G$ with no such monochromatic subgraph. Let $f(n,k,d)$ denote the minimum of $f_G(d)$ where $G$ ranges over all graphs with $n$ vertices and minimum degree at least $k$. In this paper we establish $f(n,k,d)$ whenever $k$ or $n-k$ are fixed, and $n$ is sufficiently large. We also consider the case where more than two colors are allowed.

scholarly journals Edge-Disjoint Induced Subgraphs with Given Minimum Degree

10.37236/2882 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Minimum Degree ◽  
Independent Set ◽  
Induced Subgraphs ◽  
Asymptotically Optimal ◽  
Edge Disjoint

Let $h$ be a given positive integer. For a graph with $n$ vertices and $m$ edges, what is the maximum number of pairwise edge-disjoint {\em induced} subgraphs, each having  minimum degree at least $h$? There are examples for which this number is $O(m^2/n^2)$. We prove that this bound is achievable for all graphs with polynomially many edges. For all $\epsilon > 0$, if $m \ge n^{1+\epsilon}$, then there are always $\Omega(m^2/n^2)$ pairwise edge-disjoint induced subgraphs, each having  minimum degree at least $h$. Furthermore, any two subgraphs intersect in an independent set of size at most $1+ O(n^3/m^2)$, which is shown to be asymptotically optimal.

scholarly journals Two sufficient conditions for fractional k-deleted graphs

2012 ◽  
Vol 20 (1) ◽  
pp. 265-274
Author(s):  
Xiangyang Lv
Keyword(s):  
Positive Integer ◽  
Minimum Degree ◽  
Independence Number ◽  
Sufficient Conditions ◽  
K Factor

Abstract Let G be a graph, and k a positive integer. A fractional k-factor is a way of assigning weights to the edges of a graph G (with all weights between 0 and 1) such that for each vertex the sum of the weights of the edges incident with that vertex is k. A graph G is a fractional k-deleted graph if G - e has a fractional k-factor for each e ∈ 2 E(G). In this paper, we obtain some sufficient conditions for graphs to be fractional k-deleted graphs in terms of their minimum degree and independence number. Furthermore, we show the results are best possible in some sense

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.

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.

Dominating a Family of Graphs with Small Connected Subgraphs

2000 ◽  
Vol 9 (4) ◽  
pp. 309-313 ◽  
Author(s):  
YAIR CARO ◽  
RAPHAEL YUSTER
Keyword(s):  
Positive Integer ◽  
Minimum Degree ◽  
Minimum Size ◽  
Dominating Sets ◽  
Asymptotic Results ◽  
Connected Dominating Sets ◽  
Vertex Set ◽  
Connected Subgraphs

Let F = {G1, …, Gt} be a family of n-vertex graphs defined on the same vertex-set V, and let k be a positive integer. A subset of vertices D ⊂ V is called an (F, k)-core if, for each v ∈ V and for each i = 1, …, t, there are at least k neighbours of v in Gi that belong to D. The subset D is called a connected (F, k)-core if the subgraph induced by D in each Gi is connected. Let δi be the minimum degree of Gi and let δ(F) = minti=1δi. Clearly, an (F, k)-core exists if and only if δ(F) [ges ] k, and a connected (F, k)-core exists if and only if δ(F) [ges ] k and each Gi is connected. Let c(k, F) and cc(k, F) be the minimum size of an (F, k)-core and a connected (F, k)-core, respectively. The following asymptotic results are proved for every t < ln ln δ and k < √lnδ:formula hereThe results are asymptotically tight for infinitely many families F. The results unify and extend related results on dominating sets, strong dominating sets and connected dominating sets.

scholarly journals Tiling Tripartite Graphs with $3$-Colorable Graphs

10.37236/198 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Ryan Martin ◽  
Yi Zhao
Keyword(s):  
Positive Integer ◽  
Positive Real Number ◽  
Minimum Degree ◽  
Discrete Math ◽  
The Other ◽  
Sufficient Condition ◽  
Positive Real ◽  
Tripartite Graph ◽  
Colorable Graph ◽  
Tripartite Graphs

For any positive real number $\gamma$ and any positive integer $h$, there is $N_0$ such that the following holds. Let $N\ge N_0$ be such that $N$ is divisible by $h$. If $G$ is a tripartite graph with $N$ vertices in each vertex class such that every vertex is adjacent to at least $(2/3+ \gamma) N$ vertices in each of the other classes, then $G$ can be tiled perfectly by copies of $K_{h,h,h}$. This extends the work in [Discrete Math. 254 (2002), 289–308] and also gives a sufficient condition for tiling by any fixed 3-colorable graph. Furthermore, we show that the minimum-degree $(2/3+ \gamma) N$ in our result cannot be replaced by $2N/3+ h-2$.

scholarly journals Towards a Weighted Version of the Hajnal–Szemerédi Theorem

2013 ◽  
Vol 22 (3) ◽  
pp. 346-350 ◽  
Author(s):  
JOZSEF BALOGH ◽  
GRAEME KEMKES ◽  
CHOONGBUM LEE ◽  
STEPHEN J. YOUNG
Keyword(s):  
Positive Integer ◽  
Minimum Degree ◽  
Edge Weight ◽  
Main Question ◽  
Weighted Graphs ◽  
Lower And Upper Bounds ◽  
Weighted Degree ◽  
Weighted Version ◽  
Vertex Disjoint ◽  
The Given

For a positive integer r ≥ 2, a Kr-factor of a graph is a collection vertex-disjoint copies of Kr which covers all the vertices of the given graph. The celebrated theorem of Hajnal and Szemerédi asserts that every graph on n vertices with minimum degree at least $(1-\frac{1}{r})n contains a Kr-factor. In this note, we propose investigating the relation between minimum degree and existence of perfect Kr-packing for edge-weighted graphs. The main question we study is the following. Suppose that a positive integer r ≥ 2 and a real t ∈ [0, 1] is given. What is the minimum weighted degree of Kn that guarantees the existence of a Kr-factor such that every factor has total edge weight at least $$t\binom{r}{2}$?$ We provide some lower and upper bounds and make a conjecture on the asymptotics of the threshold as n goes to infinity.

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 Covering a Graph with Cycles of Length at Least 4

10.37236/4099 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Hong Wang
Keyword(s):  
Positive Integer ◽  
Minimum Degree ◽  
Disjoint Cycles ◽  
Vertex Disjoint

Let $G$ be a graph of order $n\geq 4k$, where $k$ is a positive integer. Suppose that the minimum degree of $G$ is at least $\lceil n/2\rceil$. We show that $G$ contains $k$ vertex-disjoint cycles covering all the vertices of $G$ such that $k-1$ of them are quadrilaterals.

scholarly journals Connectivity of Fibonacci cubes, Lucas cubes and generalized cubes

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Jernej Azarija ◽  
Sandi Klavžar ◽  
Jaehun Lee ◽  
Yoomi Rho
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Minimum Degree ◽  
Edge Connectivity ◽  
Binary Word ◽  
Fibonacci Cube ◽  
International Audience ◽  
Lucas Cube

Graph Theory International audience If f is a binary word and d a positive integer, then the generalized Fibonacci cube Qd(f) is the graph obtained from the d-cube Qd by removing all the vertices that contain f as a factor, while the generalized Lucas cube Qd(lucas(f)) is the graph obtained from Qd by removing all the vertices that have a circulation containing f as a factor. The Fibonacci cube Γd and the Lucas cube Λd are the graphs Qd(11) and Qd(lucas(11)), respectively. It is proved that the connectivity and the edge-connectivity of Γd as well as of Λd are equal to ⌊ d+2 / 3⌋. Connected generalized Lucas cubes are characterized and generalized Fibonacci cubes are proved to be 2-connected. It is asked whether the connectivity equals minimum degree also for all generalized Fibonacci/Lucas cubes. It was checked by computer that the answer is positive for all f and all d≤9.

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