scholarly journals The Radio Number for Some Classes of the Cartesian Products of Complete Graphs and Cycles

2021 ◽  
Vol 1850 (1) ◽  
pp. 012014
Author(s):  
P. K. Niranjan ◽  
Srinivasa Rao Kola
Keyword(s):  
Complete Graphs ◽  
Cartesian Products ◽  
Radio Number

scholarly journals The distinguishing number of Cartesian products of complete graphs

2008 ◽  
Vol 29 (4) ◽  
pp. 922-929 ◽  
Author(s):  
Wilfried Imrich ◽  
Janja Jerebic ◽  
Sandi Klavžar
Keyword(s):  
Complete Graphs ◽  
Cartesian Products ◽  
Distinguishing Number

Proper Coloring Distance in Edge-Colored Cartesian Products of Complete Graphs and Cycles

2019 ◽  
Vol 29 (04) ◽  
pp. 1950016
Author(s):  
Ajay Arora ◽  
Eddie Cheng ◽  
Colton Magnant
Keyword(s):  
Connected Graph ◽  
The Other ◽  
Complete Graphs ◽  
Specific Class ◽  
Proper Coloring ◽  
Cartesian Products ◽  
General Graph ◽  
Proper Distance ◽  
Cyclic Graphs

An path that is edge-colored is called proper if no two consecutive edges receive the same color. A general graph that is edge-colored is called properly connected if, for every pair of vertices in the graph, there exists a properly colored path from one to the other. Given two vertices u and v in a properly connected graph G, the proper distance is the length of the shortest properly colored path from u to v. By considering a specific class of colorings that are properly connected for Cartesian products of complete and cyclic graphs, we present results on the proper distance between all pairs of vertices in the graph.

scholarly journals Distinct Distances in Graph Drawings

10.37236/831 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Paz Carmi ◽  
Vida Dujmović ◽  
Pat Morin ◽  
David R. Wood
Keyword(s):  
Maximum Degree ◽  
Combinatorial Geometry ◽  
Complete Graphs ◽  
Bounded Degree ◽  
Cartesian Products ◽  
Bounded Treewidth ◽  
Line Drawings ◽  
Straight Line ◽  
Minimum Number ◽  
Complete Bipartite Graphs

The distance-number of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the distance-number of trees, graphs with no $K^-_4$-minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distance-number of graphs with bounded degree. We prove that $n$-vertex graphs with bounded maximum degree and bounded treewidth have distance-number in ${\cal O}(\log n)$. To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth $2$ and polynomial distance-number. Similarly, we prove that there exist graphs with maximum degree $5$ and arbitrarily large distance-number. Moreover, as $\Delta$ increases the existential lower bound on the distance-number of $\Delta$-regular graphs tends to $\Omega(n^{0.864138})$.

scholarly journals Labeling amalgamations of Cartesian products of complete graphs with a condition at distance two

2014 ◽  
Vol 178 ◽  
pp. 101-108 ◽  
Author(s):  
Nathaniel Karst ◽  
Jessica Oehrlein ◽  
Denise Sakai Troxell ◽  
Junjie Zhu
Keyword(s):  
Complete Graphs ◽  
Cartesian Products

scholarly journals Cartesian Products of Some Regular Graphs Admitting Antimagic Labeling for Arbitrary Sets of Real Numbers

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Yi-Wu Chang ◽  
Shan-Pang Liu
Keyword(s):  
Complete Graphs ◽  
Regular Graphs ◽  
Real Numbers ◽  
Cartesian Products ◽  
Antimagic Labeling ◽  
Edge Labeling

An edge labeling of graph G with labels in A is an injection from E G to A , where E G is the edge set of G , and A is a subset of ℝ . A graph G is called ℝ -antimagic if for each subset A of ℝ with A = E G , there is an edge labeling with labels in A such that the sums of the labels assigned to edges incident to distinct vertices are different. The main result of this paper is that the Cartesian products of complete graphs (except K 1 ) and cycles are ℝ -antimagic.

scholarly journals Linkedness of Cartesian products of complete graphs

2021 ◽  
Author(s):  
Leif K. Jørgensen ◽  
Guillermo Pineda-Villavicencio ◽  
Julien Ugon
Keyword(s):  
Complete Graphs ◽  
Cartesian Products

Distinguishing chromatic numbers of complements of Cartesian products of complete graphs

2018 ◽  
Vol 341 (9) ◽  
pp. 2431-2441
Author(s):  
M.S. Cavers ◽  
K. Seyffarth ◽  
E.P. White
Keyword(s):  
Complete Graphs ◽  
Chromatic Numbers ◽  
Cartesian Products

Lineark-Arboricity in Product Networks

2016 ◽  
Vol 16 (03n04) ◽  
pp. 1650008
Author(s):  
YAPING MAO ◽  
ZHIWEI GUO ◽  
NAN JIA ◽  
HE LI
Keyword(s):  
Upper And Lower Bounds ◽  
Complete Graphs ◽  
Lexicographic Product ◽  
Dimensional Torus ◽  
Grid Graph ◽  
Cartesian Products ◽  
Strong Product ◽  
Multipartite Graphs ◽  
Complete Multipartite ◽  
Generalized Hypercube

A linear k-forest is a forest whose components are paths of length at most k. The linear k-arboricity of a graph G, denoted bylak(G), is the least number of linear k-forests needed to decompose G. Recently, Zuo, He, and Xue studied the exact values of the linear(n−1)-arboricity of Cartesian products of various combinations of complete graphs, cycles, complete multipartite graphs. In this paper, for general k we show thatmax{lak(G),lal(H)}≤lamax{k,l}(G□H)≤lak(G)+lal(H)for any two graphs G and H. Denote byG∘H, G×HandG⊠Hthe lexicographic product, direct product and strong product of two graphs G and H, respectively. For any two graphs G and H, we also derive upper and lower bounds oflak(G∘H),lak(G×H)andlak(G⊠H)in this paper. The linear k-arboricity of a 2-dimensional grid graph, a r-dimensional mesh, a r-dimensional torus, a r-dimensional generalized hypercube and a hyper Petersen network are also studied.

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