scholarly journals On the Gonality of Cartesian Products of Graphs

10.37236/9307 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Ivan Aidun ◽  
Ralph Morrison
Keyword(s):  
Upper Bound ◽  
Algebraic Curve ◽  
Cartesian Product ◽  
Metric Graphs ◽  
Systematic Treatment ◽  
Cartesian Products ◽  
Product Graphs ◽  
Chip Firing ◽  
Cartesian Products Of Graphs

In this paper we provide the first systematic treatment of Cartesian products of graphs and their divisorial gonality, which is a tropical version of the gonality of an algebraic curve defined in terms of chip-firing.  We prove an upper bound on the gonality of the Cartesian product of any two graphs, and determine instances where this bound holds with equality, including for the $m\times n$ rook's graph with $\min\{m,n\}\leq 5$.  We use our upper bound to prove that Baker's gonality conjecture holds for the Cartesian product of any two graphs with two or more vertices each, and we determine precisely which nontrivial product graphs have gonality equal to Baker's conjectural upper bound.  We also extend some of our results to metric graphs.

scholarly journals Strong geodetic problem on Cartesian products of graphs

2018 ◽  
Vol 52 (1) ◽  
pp. 205-216 ◽  
Author(s):  
Vesna Iršič ◽  
Sandi Klavžar
Keyword(s):  
Shortest Path ◽  
Upper Bound ◽  
Cartesian Product ◽  
Cartesian Products ◽  
Cartesian Product Of Graphs ◽  
Geodetic Number ◽  
Recent Variation ◽  
Product Of Graphs ◽  
Strong Geodetic Number ◽  
Cartesian Products Of Graphs

The strong geodetic problem is a recent variation of the geodetic problem. For a graph G, its strong geodetic number sg(G) is the cardinality of a smallest vertex subset S, such that each vertex of G lies on a fixed shortest path between a pair of vertices from S. In this paper, the strong geodetic problem is studied on the Cartesian product of graphs. A general upper bound for sg(G □ H) is determined, as well as exact values for Km □ Kn, K1,k □ Pl, and prisms over Kn–e. Connections between the strong geodetic number of a graph and its subgraphs are also discussed.

scholarly journals Peg Solitaire on Cartesian Products of Graphs

2021 ◽  
Vol 37 (3) ◽  
pp. 907-917
Author(s):  
Martin Kreh ◽  
Jan-Hendrik de Wiljes
Keyword(s):  
Cartesian Product ◽  
Cartesian Products ◽  
Connected Graphs ◽  
Grid Graphs ◽  
Cartesian Products Of Graphs

AbstractIn 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. In the same article, the authors proved some results on the solvability of Cartesian products, given solvable or distance 2-solvable graphs. We extend these results to Cartesian products of certain unsolvable graphs. In particular, we prove that ladders and grid graphs are solvable and, further, even the Cartesian product of two stars, which in a sense are the “most” unsolvable graphs.

scholarly journals A NOTE ON EDGE-CONNECTIVITY OF THE CARTESIAN PRODUCT OF GRAPHS

2011 ◽  
Vol 84 (1) ◽  
pp. 171-176
Author(s):  
LAKOA FITINA ◽  
C. T. LENARD ◽  
T. M. MILLS
Keyword(s):  
Cartesian Product ◽  
Edge Connectivity ◽  
Cartesian Products ◽  
Cartesian Product Of Graphs ◽  
Necessary And Sufficient ◽  
Product Of Graphs ◽  
Cartesian Products Of Graphs

AbstractThe main aim of this paper is to establish conditions that are necessary and sufficient for the edge-connectivity of the Cartesian product of two graphs to equal the sum of the edge-connectivities of the factors. The paper also clarifies an issue that has arisen in the literature on Cartesian products of graphs.

scholarly journals ON THE CROSSING NUMBER OF THE CARTESIAN PRODUCT OF A SUNLET GRAPH AND A STAR GRAPH

2019 ◽  
Vol 100 (1) ◽  
pp. 5-12
Author(s):  
MICHAEL HAYTHORPE ◽  
ALEX NEWCOMBE
Keyword(s):  
Upper Bound ◽  
Cartesian Product ◽  
Crossing Number ◽  
Star Graph ◽  
Cartesian Products ◽  
Crossing Numbers ◽  
First Time

The exact crossing number is only known for a small number of families of graphs. Many of the families for which crossing numbers have been determined correspond to cartesian products of two graphs. Here, the cartesian product of the sunlet graph, denoted ${\mathcal{S}}_{n}$, and the star graph, denoted $K_{1,m}$, is considered for the first time. It is proved that the crossing number of ${\mathcal{S}}_{n}\Box K_{1,2}$ is $n$, and the crossing number of ${\mathcal{S}}_{n}\Box K_{1,3}$ is $3n$. An upper bound for the crossing number of ${\mathcal{S}}_{n}\Box K_{1,m}$ is also given.

scholarly journals Connected domination game played on Cartesian products

2019 ◽  
Vol 17 (1) ◽  
pp. 1269-1280 ◽  
Author(s):  
Csilla Bujtás ◽  
Pakanun Dokyeesun ◽  
Vesna Iršič ◽  
Sandi Klavžar
Keyword(s):  
Lower Bound ◽  
Domination Number ◽  
Cartesian Product ◽  
Connected Subgraph ◽  
Arbitrary Graph ◽  
Cartesian Products ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Game Domination Number ◽  
Domination Game

Abstract The connected domination game on a graph G is played by Dominator and Staller according to the rules of the standard domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of G. If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the connected game domination number of G. Here this invariant is studied on Cartesian product graphs. A general upper bound is proved and demonstrated to be sharp on Cartesian products of stars with paths or cycles. The connected game domination number is determined for Cartesian products of P3 with arbitrary paths or cycles, as well as for Cartesian products of an arbitrary graph with Kk for the cases when k is relatively large. A monotonicity theorem is proved for products with one complete factor. A sharp general lower bound on the connected game domination number of Cartesian products is also established.

Optimal Diffusion Schemes and Load Balancing on Product Graphs

2004 ◽  
Vol 14 (01) ◽  
pp. 61-73 ◽  
Author(s):  
ROBERT ELSÄSSER ◽  
BURKHARD MONIEN ◽  
ROBERT PREIS ◽  
ANDREAS FROMMER
Keyword(s):  
Load Balancing ◽  
Nearest Neighbor ◽  
Cartesian Product ◽  
Cartesian Products ◽  
Optimal Scheme ◽  
Alternating Direction ◽  
Product Graphs ◽  
Diffusion Scheme ◽  
General Graphs ◽  
Number Of Iterations

We discuss nearest neighbor load balancing schemes on processor networks which are represented by a cartesian product of graphs and present a new optimal diffusion scheme for general graphs. In the first part of the paper, we introduce the Alternating-Direction load balancing scheme, which reduces the number of load balance iterations by a factor of 2 for cartesian products of graphs. The resulting flow is theoretically analyzed and can be very high for certain cases. Therefore, we further present the Mixed-Direction scheme which needs the same number of iterations but computes in most cases a much smaller flow. In the second part of the paper, we present a simple optimal diffusion scheme for general graphs, calculating a balancing flow which is minimal in the l2 norm. It is based on the spectra of the graph representing the network and needs only m-1 iterations to balance the load with m being the number of distinct eigenvalues. Known optimal diffusion schemes have the same performance, however the optimal scheme presented in this paper can be implemented in a very simple manner. The number of iterations of optimal diffusion schemes is independent of the load scenario and, thus, they are practical for networks which represent graphs with known spectra. Finally, our experiments exhibit that the new optimal scheme can successfully be combined with the Alternating-Direction and Mixed-Direction schemes for efficient load balancing on product graphs.

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 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.

scholarly journals Separation of Cartesian products of graphs into several connected components by the removal of edges

2021 ◽  
pp. 18-18
Author(s):  
Simon Spacapan
Keyword(s):  
Cartesian Product ◽  
Connected Components ◽  
Edge Connectivity ◽  
Cartesian Products ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Cartesian Products Of Graphs

Let G = (V (G),E(G)) be a graph. A set S ? E(G) is an edge k-cut in G if the graph G-S = (V (G), E(G) \ S) has at least k connected components. The generalized k-edge connectivity of a graph G, denoted as ?k(G), is the minimum cardinality of an edge k-cut in G. In this article we determine generalized 3-edge connectivity of Cartesian product of connected graphs G and H and describe the structure of any minimum edge 3-cut in G2H. The generalized 3-edge connectivity ?3(G2H) is given in terms of ?3(G) and ?3(H) and in terms of other invariants of factors G and H.

scholarly journals Game Chromatic Number of Cartesian Product Graphs

10.37236/796 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
T. Bartnicki ◽  
B. Brešar ◽  
J. Grytczuk ◽  
M. Kovše ◽  
Z. Miechowicz ◽  
...  
Keyword(s):  
Chromatic Number ◽  
Cartesian Product ◽  
Chromatic Numbers ◽  
Cartesian Products ◽  
Game Chromatic Number ◽  
Coloring Number ◽  
Product Graphs ◽  
Complete Bipartite Graphs ◽  
Game Coloring Number ◽  
Product Of Graphs

The game chromatic number $\chi _{g}$ is considered for the Cartesian product $G\,\square \,H$ of two graphs $G$ and $H$. Exact values of $\chi _{g}(K_2\square H)$ are determined when $H$ is a path, a cycle, or a complete graph. By using a newly introduced "game of combinations" we show that the game chromatic number is not bounded in the class of Cartesian products of two complete bipartite graphs. This result implies that the game chromatic number $\chi_{g}(G\square H)$ is not bounded from above by a function of game chromatic numbers of graphs $G$ and $H$. An analogous result is derived for the game coloring number of the Cartesian product of graphs.

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