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.

On the b-chromatic number of some graph products

2012 ◽  
Vol 49 (2) ◽  
pp. 156-169 ◽  
Author(s):  
Marko Jakovac ◽  
Iztok Peterin
Keyword(s):  
Chromatic Number ◽  
Upper And Lower Bounds ◽  
Vertex Coloring ◽  
Graph Products ◽  
Lexicographic Product ◽  
Arbitrary Graph ◽  
Strong Product ◽  
Color Class ◽  
Complete Bipartite Graphs ◽  
Proper Vertex

A b-coloring is a proper vertex coloring of a graph such that each color class contains a vertex that has a neighbor in all other color classes and the b-chromatic number is the largest integer φ(G) for which a graph has a b-coloring with φ(G) colors. We determine some upper and lower bounds for the b-chromatic number of the strong product G ⊠ H, the lexicographic product G[H] and the direct product G × H and give some exact values for products of paths, cycles, stars, and complete bipartite graphs. We also show that the b-chromatic number of Pn ⊠ H, Cn ⊠ H, Pn[H], Cn[H], and Km,n[H] can be determined for an arbitrary graph H, when integers m and n are large enough.

Decompositions of some regular graphs into unicyclic graphs of order five

2019 ◽  
Vol 11 (04) ◽  
pp. 1950042 ◽  
Author(s):  
P. Paulraja ◽  
T. Sivakaran
Keyword(s):  
Necessary Conditions ◽  
Discrete Math ◽  
Complete Graphs ◽  
Regular Graphs ◽  
Necessary And Sufficient Condition ◽  
Unicyclic Graphs ◽  
Multipartite Graphs ◽  
Complete Multipartite ◽  
Necessary And Sufficient ◽  
Product Of Graphs

For a graph [Formula: see text] and a subgraph [Formula: see text] of [Formula: see text] an [Formula: see text]-decomposition of [Formula: see text] is a partition of the edge set of [Formula: see text] into subsets [Formula: see text] [Formula: see text] such that each [Formula: see text] induces a graph isomorphic to [Formula: see text] It is proved that the necessary conditions are sufficient for the existence of an [Formula: see text]-decomposition of the graph [Formula: see text] where [Formula: see text] is any simple connected unicyclic graph of order five, × denotes the tensor product of graphs and [Formula: see text] denotes the multiplicity of the edges. In fact, using the above characterization, a necessary and sufficient condition for the graph [Formula: see text] [Formula: see text] and [Formula: see text] to admit an [Formula: see text]-decomposition is obtained. Similar results for the complete graphs and complete multipartite graphs are proved in: [J.-C. Bermond et al. [Formula: see text]-decomposition of [Formula: see text], where [Formula: see text] has four vertices or less, Discrete Math. 19 (1977) 113–120, J.-C. Bermond et al. Decomposition of complete graphs into isomorphic subgraphs with five verices, Ars Combin. 10 (1980) 211–254, M. H. Huang, Decomposing complete equipartite graphs into connected unicyclic graphs of size five, Util. Math. 97 (2015) 109–117].

scholarly journals Line Graphs and Middle Graphs that are Divisor Graphs

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Line Graphs ◽  
Complete Graphs ◽  
Permutation Graphs ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Complete Multipartite ◽  
Cycle Permutation

In this paper it is determined when the line graphs and the middle graphs of some classes of graphs are divisor graphs. Complete characterizations for cycles, trees, complete graphs and complete multipartite graphs whose line graphs (middle graphs) are divisor graphs are obtained. It is also shown that the line graphs and the middle graphs of the cycle permutation graphs are never divisor graphs.

On mixing in continuous-time quantum walks on some circulant graphs

2003 ◽  
Vol 3 (6) ◽  
pp. 611-618
Author(s):  
A. Ahmadi ◽  
R. Belk ◽  
C. Tamon ◽  
C. Wendler
Keyword(s):  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Complete Graphs ◽  
Circulant Graphs ◽  
Multipartite Graphs ◽  
Time Quantum ◽  
Complete Multipartite ◽  
Cycle Graph ◽  
Mixing Property

Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that the continuous-time quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs that have the instantaneous exactly uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs of the symmetric group lack this mixing property as well.

Pseudo orthogonal Latin squares

2021 ◽  
Vol 31 (1) ◽  
pp. 5-17
Author(s):  
Shahab Faruqi ◽  
S. A. Katre ◽  
Manisha Garg
Keyword(s):  
Tensor Product ◽  
Latin Squares ◽  
Complete Graphs ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Complete Multipartite

Abstract Two Latin squares A, B of order n are called pseudo orthogonal if for any 1 ≤ i, j ≤ n there exists a k, 1 ≤ k ≤ n, such that A(i, k) = B(j, k). We prove that the existence of a family of m mutually pseudo orthogonal Latin squares of order n is equivalent to the existence of a family of m mutually orthogonal Latin squares of order n. We also obtain exact values of clique partition numbers of several classes of complete multipartite graphs and of the tensor product of complete graphs.

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