The (3, l)-Rainbow Edge-Index of Cartesian Product Graphs

2017 ◽  
Vol 17 (03n04) ◽  
pp. 1741009
Author(s):  
YUEFANG SUN
Keyword(s):  
Upper Bound ◽  
Cartesian Product ◽  
Edge Coloring ◽  
Index Formula ◽  
Colored Graph ◽  
Common Edge ◽  
Edge Connectivity ◽  
Edge Disjoint ◽  
Product Graphs ◽  
Cartesian Product Graphs

For a graph G and a vertex subset [Formula: see text] of at least two vertices, an S-tree is a subgraph T of G that is a tree with [Formula: see text]. Two S-trees are said to be edge-disjoint if they have no common edge. Let [Formula: see text] denote the maximum number of edge-disjoint S-trees in G. For an integer K with [Formula: see text], the generalized k-edge-connectivity is defined as [Formula: see text]. An S-tree in an edge-colored graph is rainbow if no two edges of it are assigned the same color. Let [Formula: see text] and l be integers with [Formula: see text], the [Formula: see text]-rainbow edge-index [Formula: see text] of G is the smallest number of colors needed in an edge-coloring of G such that for every set S of k vertices of G, there exist l edge-disjoint rainbow S-trees.In this paper, we study the [Formula: see text]-rainbow edge-index of Cartesian product graphs and get a sharp upper bound for [Formula: see text] , where G and H are connected graphs with orders at least three, and [Formula: see text] denotes the Cartesian product of G and H.

scholarly journals On super edge-connectivity of Cartesian product graphs

Networks ◽  
2007 ◽  
Vol 49 (2) ◽  
pp. 152-157 ◽  
Author(s):  
Min Lü ◽  
Guo-Liang Chen ◽  
Jun-Ming Xu
Keyword(s):  
Cartesian Product ◽  
Edge Connectivity ◽  
Product Graphs ◽  
Cartesian Product Graphs

scholarly journals Strong Geodetic Number in Some Networks

2019 ◽  
Vol 11 (2) ◽  
pp. 20
Author(s):  
Huifen Ge ◽  
Zhao Wang ◽  
Jinyu Zou
Keyword(s):  
Upper Bound ◽  
Cartesian Product ◽  
Geodetic Number ◽  
Product Graphs ◽  
Geodetic Set ◽  
Cartesian Product Graphs ◽  
Strong Geodetic Number

A vertex subset S of a graph is called a strong geodetic set if there exists a choice of exactly one geodesic for each pair of vertices of S in such a way that these (|S| 2) geodesics cover all the vertices of graph G. The strong geodetic number of G, denoted by sg(G), is the smallest cardinality of a strong geodetic set. In this paper, we give an upper bound of strong geodetic number of the Cartesian product graphs and study this parameter for some Cartesian product networks.

ON THE EDGE-CONNECTIVITY OF CARTESIAN PRODUCT GRAPHS

2008 ◽  
Vol 01 (01) ◽  
pp. 93-98 ◽  
Author(s):  
Sandi Klavžar ◽  
Simon Špacapan
Keyword(s):  
Short Proof ◽  
Minimum Degree ◽  
Cartesian Product ◽  
Edge Connectivity ◽  
Closed Expression ◽  
Cartesian Power ◽  
Product Graphs ◽  
Cartesian Product Graphs

A short proof for the closed expression of the edge-connectivity of Cartesian product graphs is given and the structure of minimum edge cuts is described. It is also proved that the connectivity and edge-connectivity of an arbitrary Cartesian power equals its minimum degree.

scholarly journals Chromatic Number for a Generalization of Cartesian Product Graphs

10.37236/160 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Daniel Král' ◽  
Douglas B. West
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Cartesian Product ◽  
Rectangular Grid ◽  
Product Graphs ◽  
Cartesian Product Graphs

Let ${\cal G}$ be a class of graphs. A $d$-fold grid over ${\cal G}$ is a graph obtained from a $d$-dimensional rectangular grid of vertices by placing a graph from ${\cal G}$ on each of the lines parallel to one of the axes. Thus each vertex belongs to $d$ of these subgraphs. The class of $d$-fold grids over ${\cal G}$ is denoted by ${\cal G}^d$. Let $f({\cal G};d)=\max_{G\in{\cal G}^d}\chi(G)$. If each graph in ${\cal G}$ is $k$-colorable, then $f({\cal G};d)\le k^d$. We show that this bound is best possible by proving that $f({\cal G};d)=k^d$ when ${\cal G}$ is the class of all $k$-colorable graphs. We also show that $f({\cal G};d)\ge{\left\lfloor\sqrt{{d\over 6\log d}}\right\rfloor}$ when ${\cal G}$ is the class of graphs with at most one edge, and $f({\cal G};d)\ge {\left\lfloor{d\over 6\log d}\right\rfloor}$ when ${\cal G}$ is the class of graphs with maximum degree $1$.

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