scholarly journals Minimum degree of minimal defect n-extendable bipartite graphs

2009 ◽  
Vol 309 (21) ◽  
pp. 6255-6264 ◽  
Author(s):  
Xuelian Wen ◽  
Zihui Yang
Keyword(s):  
Minimum Degree ◽  
Bipartite Graphs ◽  
Minimal Defect

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 Almost well-dominated bipartite graphs with minimum degree at least two

2020 ◽  
Author(s):  
Hadi Alizadeh ◽  
Didem Gözüpek
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Maximum Cardinality ◽  
Minimum Cardinality ◽  
Upper Domination Number ◽  
The Difference ◽  
Two Parameters

A dominating set in a graph $G=(V,E)$ is a set $S$ such that every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. While the minimum cardinality of a dominating set in $G$ is called the domination number of $G$ denoted by $\gamma(G)$, the maximum cardinality of a minimal dominating set in $G$ is called the upper domination number of $G$ denoted by $\Gamma(G)$. We call the difference between these two parameters the \textit{domination gap} of $G$ and denote it by $\mu_d(G) = \Gamma(G) - \gamma(G)$. While a graph $G$ with $\mu_d(G)=0$ is said to be a \textit{well-dominated} graph, we call a graph $G$ with $\mu_d(G)=1$ an \textit{almost well-dominated} graph. In this work, we first establish an upper bound for the cardinality of bipartite graphs with $\mu_d(G)=k$, where $k\geq1$, and minimum degree at least two. We then provide a complete structural characterization of almost well-dominated bipartite graphs with minimum degree at least two. While the results by Finbow et al.~\cite{domination} imply that a 4-cycle is the only well-dominated bipartite graph with minimum degree at least two, we prove in this paper that there exist precisely 31 almost well-dominated bipartite graphs with minimum degree at least two.

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 Existence of Spanning ℱ-Free Subgraphs with Large Minimum Degree

2016 ◽  
Vol 26 (3) ◽  
pp. 448-467
Author(s):  
G. PERARNAU ◽  
B. REED
Keyword(s):  
Regular Graph ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Complete Bipartite Graphs

Let ℱ be a family of graphs and letdbe large enough. For everyd-regular graphG, we study the existence of a spanning ℱ-free subgraph ofGwith large minimum degree. This problem is well understood if ℱ does not contain bipartite graphs. Here we provide asymptotically tight results for many families of bipartite graphs such as cycles or complete bipartite graphs. To prove these results, we study a locally injective analogue of the question.

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.

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 Large Monochromatic Components in Almost Complete Graphs and Bipartite Graphs

10.37236/9824 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Zoltán Füredi ◽  
Ruth Luo
Keyword(s):  
Minimum Degree ◽  
Edge Coloring ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Connected Component ◽  
Vertex Graph

Gyárfas proved that every coloring of the edges of $K_n$ with $t+1$ colors contains a monochromatic connected component of size at least $n/t$. Later, Gyárfás and Sárközy asked for which values of $\gamma=\gamma(t)$ does the following strengthening for almost complete graphs hold: if $G$ is an $n$-vertex graph with minimum degree at least $(1-\gamma)n$, then every $(t+1)$-edge coloring of $G$ contains a monochromatic component of size at least $n/t$. We show $\gamma= 1/(6t^3)$ suffices, improving a result of DeBiasio, Krueger, and Sárközy.

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.

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