scholarly journals A Note on the Adversary Degree Associated Reconstruction Number of Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
S. Monikandan ◽  
S. Sundar Raj ◽  
C. Jayasekaran ◽  
A. P. Santhakumaran
Keyword(s):  
Bipartite Graphs ◽  
Minimum Number ◽  
Complete Bipartite Graphs

A vertex-deleted subgraph of a graph G is called a card of G. A card of G with which the degree of the deleted vertex is also given is called a degree associated card (or dacard) of G. The degree associated reconstruction number drn (G) of a graph G is the size of the smallest collection of dacards of G that uniquely determines G. The adversary degree associated reconstruction number of a graph G, adrn(G), is the minimum number k such that every collection of k dacards of G that uniquely determines G. In this paper, we show that adrn of wheels and complete bipartite graphs on at least 4 vertices is 2 or 3.

scholarly journals Dicycle Cover of Hamiltonian Oriented Graphs

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Khalid A. Alsatami ◽  
Hong-Jian Lai ◽  
Xindong Zhang
Keyword(s):  
Bipartite Graphs ◽  
Upper Bounds ◽  
Oriented Graphs ◽  
Minimum Number ◽  
Complete Bipartite Graphs ◽  
Number Of Classes

A dicycle cover of a digraph D is a family F of dicycles of D such that each arc of D lies in at least one dicycle in F. We investigate the problem of determining the upper bounds for the minimum number of dicycles which cover all arcs in a strong digraph. Best possible upper bounds of dicycle covers are obtained in a number of classes of digraphs including strong tournaments, Hamiltonian oriented graphs, Hamiltonian oriented complete bipartite graphs, and families of possibly non-Hamiltonian digraphs obtained from these digraphs via a sequence of 2-sum operations.

scholarly journals On a Theorem of Erdős, Rubin, and Taylor on Choosability of Complete Bipartite Graphs

10.37236/1670 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Alexandr Kostochka
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Uniform Hypergraph ◽  
Ordinary Sense ◽  
Minimum Order ◽  
Minimum Number ◽  
Complete Bipartite Graphs

Erdős, Rubin, and Taylor found a nice correspondence between the minimum order of a complete bipartite graph that is not $r$-choosable and the minimum number of edges in an $r$-uniform hypergraph that is not $2$-colorable (in the ordinary sense). In this note we use their ideas to derive similar correspondences for complete $k$-partite graphs and complete $k$-uniform $k$-partite hypergraphs.

scholarly journals On acyclic edge-coloring of complete bipartite graphs

2017 ◽  
Vol 340 (3) ◽  
pp. 481-493
Author(s):  
Ayineedi Venkateswarlu ◽  
Santanu Sarkar ◽  
Sai Mali Ananthanarayanan
Keyword(s):  
Edge Coloring ◽  
Bipartite Graphs ◽  
Acyclic Edge Coloring ◽  
Complete Bipartite Graphs

scholarly journals On randomly 3-axial graphs

1982 ◽  
Vol 25 (2) ◽  
pp. 187-206
Author(s):  
Yousef Alavi ◽  
Sabra S. Anderson ◽  
Gary Chartrand ◽  
S.F. Kapoor
Keyword(s):  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Complete Graphs ◽  
Initial Vertex ◽  
Complete Bipartite Graphs

A graph G, every vertex of which has degree at least three, is randomly 3-axial if for each vertex v of G, any ordered collection of three paths in G of length one with initial vertex v can be cyclically randomly extended to produce three internally disjoint paths which contain all the vertices of G. Randomly 3-axial graphs of order p > 4 are characterized for p ≢ 1 (mod 3), and are shown to be either complete graphs or certain regular complete bipartite graphs.

ON MOSTAR INDEX OF GRAPHS

2021 ◽  
Vol 10 (4) ◽  
pp. 2115-2129
Author(s):  
P. Kandan ◽  
S. Subramanian
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Bipartite Graphs ◽  
Topological Indices ◽  
Great Success ◽  
Complete Graphs ◽  
Molecular Graphs ◽  
Graphs And Networks ◽  
Complete Bipartite Graphs ◽  
Simple Connected Graph

On the great success of bond-additive topological indices like Szeged, Padmakar-Ivan, Zagreb, and irregularity measures, yet another index, the Mostar index, has been introduced recently as a peripherality measure in molecular graphs and networks. For a connected graph G, the Mostar index is defined as $$M_{o}(G)=\displaystyle{\sum\limits_{e=gh\epsilon E(G)}}C(gh),$$ where $C(gh) \,=\,\left|n_{g}(e)-n_{h}(e)\right|$ be the contribution of edge $uv$ and $n_{g}(e)$ denotes the number of vertices of $G$ lying closer to vertex $g$ than to vertex $h$ ($n_{h}(e)$ define similarly). In this paper, we prove a general form of the results obtained by $Do\check{s}li\acute{c}$ et al.\cite{18} for compute the Mostar index to the Cartesian product of two simple connected graph. Using this result, we have derived the Cartesian product of paths, cycles, complete bipartite graphs, complete graphs and to some molecular graphs.

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