scholarly journals A SUFFICIENT CONDITION FOR A PAIR OF SEQUENCES TO BE BIPARTITE GRAPHIC

2016 ◽  
Vol 94 (2) ◽  
pp. 195-200 ◽  
Author(s):  
GRANT CAIRNS ◽  
STACEY MENDAN ◽  
YURI NIKOLAYEVSKY
Keyword(s):  
Bipartite Graph ◽  
Sufficient Condition ◽  
Degree Sequences ◽  
Integer Sequences ◽  
Finite Integer

We present a sufficient condition for a pair of finite integer sequences to be degree sequences of a bipartite graph, based only on the lengths of the sequences and their largest and smallest elements.

scholarly journals The degree sequences of a graph with restrictions

2021 ◽  
Vol 5 (2) ◽  
pp. 68
Author(s):  
Rikio Ichishima ◽  
Francesc A. Muntaner-Batle ◽  
Miquel Rius-Font ◽  
Yukio Takahashi
Keyword(s):  
Bipartite Graph ◽  
Sufficient Condition ◽  
Degree Sequences ◽  
Necessary And Sufficient Condition ◽  
Finite Sequences ◽  
Necessary And Sufficient

<p>Two finite sequences <em>s</em><sub>1 </sub>and <em>s</em><sub>2</sub> of nonnegative integers are called bigraphical if there exists a bipartite graph <em>G</em> with partite sets <em>V</em><sub>1</sub> and <em>V</em><sub>2</sub> such that <em>s</em><sub>1</sub> and <em>s</em><sub>2</sub> are the degrees in <em>G </em>of the vertices in <em>V</em><sub>1</sub> and <em>V</em><sub>2</sub>, respectively. In this paper, we introduce the concept of <em>1</em>-graphical sequences and present a necessary and sufficient condition for a sequence to be <em>1</em>-graphical in terms of bigraphical sequences.</p>

scholarly journals Another H-super magic decompositions of the lexicographic product of graphs

2018 ◽  
Vol 2 (2) ◽  
pp. 72
Author(s):  
H Hendy ◽  
Kiki A. Sugeng ◽  
A.N.M Salman ◽  
Nisa Ayunda
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Graph Decomposition ◽  
Graph Labeling ◽  
Sufficient Condition ◽  
Lexicographic Product ◽  
Simple Graphs ◽  
Product Of Graphs

<p>Let <span class="math"><em>H</em></span> and <span class="math"><em>G</em></span> be two simple graphs. The concept of an <span class="math"><em>H</em></span>-magic decomposition of <span class="math"><em>G</em></span> arises from the combination between graph decomposition and graph labeling. A decomposition of a graph <span class="math"><em>G</em></span> into isomorphic copies of a graph <span class="math"><em>H</em></span> is <span class="math"><em>H</em></span>-magic if there is a bijection <span class="math"><em>f</em> : <em>V</em>(<em>G</em>) ∪ <em>E</em>(<em>G</em>) → {1, 2, ..., ∣<em>V</em>(<em>G</em>) ∪ <em>E</em>(<em>G</em>)∣}</span> such that the sum of labels of edges and vertices of each copy of <span class="math"><em>H</em></span> in the decomposition is constant. A lexicographic product of two graphs <span class="math"><em>G</em><sub>1</sub></span> and <span class="math"><em>G</em><sub>2</sub>, </span> denoted by <span class="math"><em>G</em><sub>1</sub>[<em>G</em><sub>2</sub>], </span> is a graph which arises from <span class="math"><em>G</em><sub>1</sub></span> by replacing each vertex of <span class="math"><em>G</em><sub>1</sub></span> by a copy of the <span class="math"><em>G</em><sub>2</sub></span> and each edge of <span class="math"><em>G</em><sub>1</sub></span> by all edges of the complete bipartite graph <span class="math"><em>K</em><sub><em>n</em>, <em>n</em></sub></span> where <span class="math"><em>n</em></span> is the order of <span class="math"><em>G</em><sub>2</sub>.</span> In this paper we provide a sufficient condition for <span class="math">$\overline{C_{n}}[\overline{K_{m}}]$</span> in order to have a <span class="math">$P_{t}[\overline{K_{m}}]$</span>-magic decompositions, where <span class="math"><em>n</em> &gt; 3, <em>m</em> &gt; 1, </span> and <span class="math"><em>t</em> = 3, 4, <em>n</em> − 2</span>.</p>

scholarly journals Forcing $k$-Repetitions in Degree Sequences

10.37236/3503 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Yair Caro ◽  
Asaf Shapira ◽  
Raphael Yuster
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Main Tool ◽  
Geometric Approach ◽  
Degree Sequences ◽  
Integer Sequences ◽  
Sum Theorem ◽  
Zero Sum

One of the most basic results in graph theory states that every graph with at least two vertices has two vertices with the same degree. Since there are graphs without $3$ vertices of the same degree, it is natural to ask if for any fixed $k$, every graph $G$ is "close" to a graph $G'$ with  $k$ vertices of the same degree. Our main result in this paper is that this is indeed the case. Specifically, we show that for any positive integer $k$, there is a constant $C=C(k)$, so that given any graph $G$, one can remove from $G$ at most $C$ vertices and thus obtain a new graph $G'$ that contains at least $\min\{k,|G|-C\}$ vertices of the same degree.Our main tool is a multidimensional zero-sum theorem for integer sequences, which we prove using an old geometric approach of Alon and Berman.

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 On Uniformly Distributed Dilates of Finite Integer Sequences

2000 ◽  
Vol 82 (2) ◽  
pp. 165-187
Author(s):  
S.V Konyagin ◽  
I.Z Ruzsa ◽  
W Schlag
Keyword(s):  
Integer Sequences ◽  
Finite Integer

A sufficient condition for a bipartite graph to have ak-factor

1988 ◽  
Vol 12 (1) ◽  
pp. 141-151 ◽  
Author(s):  
Hikoe Enomoto ◽  
Katsuhiro Ota ◽  
Mikio Kano
Keyword(s):  
Bipartite Graph ◽  
Sufficient Condition

scholarly journals Some Properties of Regular Line Graphs

2008 ◽  
pp. 44-49
Keyword(s):  
Key Words ◽  
Bipartite Graph ◽  
Line Graph ◽  
Bipartite Graphs ◽  
Line Graphs ◽  
Sufficient Condition ◽  
Regular Graphs ◽  
Necessary And Sufficient Condition ◽  
Regular Line ◽  
Necessary And Sufficient

In this paper, the concept of regular line graph has been introduced. The maximum number of vertices with different degrees in the regular line graphs has also been studied. Further, the necessary and sufficient condition for regular line graph to be bipartite graph have also been proved. Key words: Line Graphs, Regular graphs, Connected graphs, Bipartite Graphs.

scholarly journals Sufficient Condition and Algorithm for Hamiltonian in 3-Connected 3-Regular Planar Bipartite Graph

2015 ◽  
Vol 117 (11) ◽  
pp. 6-10
Author(s):  
Md. Khaliluzzaman ◽  
Md. Monirul Islam ◽  
Md. Monjur Hasan
Keyword(s):  
Bipartite Graph ◽  
Sufficient Condition

scholarly journals Graph Powers and Graph Homomorphisms

10.37236/289 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Hossein Hajiabolhassan ◽  
Ali Taherkhani
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Rational Number ◽  
Complete Graph ◽  
Rational Numbers ◽  
Sufficient Condition ◽  
Fractional Powers ◽  
Circular Chromatic Number ◽  
Equivalent Definition ◽  
Graph Homomorphisms

In this paper, we investigate some basic properties of fractional powers. In this regard, we show that for any non-bipartite graph $G$ and positive rational numbers ${2r+1\over 2s+1} < {2p+1\over 2q+1}$, we have $G^{2r+1\over 2s+1} < G^{2p+1\over 2q+1}$. Next, we study the power thickness of $G$, that is, the supremum of rational numbers ${2r+1\over 2s+1}$ such that $G$ and $G^{2r+1\over 2s+1}$ have the same chromatic number. We prove that the power thickness of any non-complete circular complete graph is greater than one. This provides a sufficient condition for the equality of the chromatic number and the circular chromatic number of graphs. Finally, we introduce an equivalent definition for the circular chromatic number of graphs in terms of fractional powers. Also, we show that for any non-bipartite graph $G$ if $0 < {{2r+1}\over {2s+1}} \leq {{\chi(G)}\over{3(\chi(G)-2)}}$, then $\chi(G^{{2r+1}\over {2s+1}})=3$. Moreover, $\chi(G)\neq\chi_c(G)$ if and only if there exists a rational number ${{2r+1}\over {2s+1}}>{{\chi(G)}\over{3(\chi(G)-2)}}$ for which $\chi(G^{{2r+1}\over {2s+1}})= 3$.

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