Distinguishing Chromatic Number of Cartesian Products of Graphs

2010 ◽  
Vol 24 (1) ◽  
pp. 82-100 ◽  
Author(s):  
Jeong Ok Choi ◽  
Stephen G. Hartke ◽  
Hemanshu Kaul
Keyword(s):  
Chromatic Number ◽  
Cartesian Products ◽  
Cartesian Products Of Graphs

scholarly journals On the Alon-Tarsi Number and Chromatic-Choosability of Cartesian Products of Graphs

10.37236/7740 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Hemanshu Kaul ◽  
Jeffrey A. Mudrock
Keyword(s):  
Chromatic Number ◽  
Cartesian Product ◽  
Hamiltonian Path ◽  
Complete Graphs ◽  
Cartesian Products ◽  
Hamilton Path ◽  
Odd Cycle ◽  
Delta G ◽  
List Chromatic Number ◽  
Cartesian Products Of Graphs

We study the list chromatic number of Cartesian products of graphs through the Alon-Tarsi number as defined by Jensen and Toft (1995) in their seminal book on graph coloring problems. The Alon-Tarsi number of $G$, $AT(G)$, is the smallest $k$ for which there is an orientation, $D$, of $G$ with max indegree $k\!-\!1$ such that the number of even and odd circulations contained in $D$ are different. It is known that $\chi(G) \leq \chi_\ell(G) \leq \chi_p(G) \leq AT(G)$, where  $\chi(G)$ is the chromatic number, $\chi_\ell(G)$ is the list chromatic number, and $\chi_p(G)$ is the paint number of $G$. In this paper we find families of graphs $G$ and $H$ such that $\chi(G \square H) = AT(G \square H)$, reducing this sequence of inequalities to equality. We show that the Alon-Tarsi number of the Cartesian product of an odd cycle and a path is always equal to 3. This result is then extended to show that if $G$ is an odd cycle or a complete graph and $H$ is a graph on at least two vertices containing the Hamilton path $w_1, w_2, \ldots, w_n$ such that for each $i$, $w_i$ has a most $k$ neighbors among $w_1, w_2, \ldots, w_{i-1}$, then $AT(G \square H) \leq \Delta(G)+k$ where $\Delta(G)$ is the maximum degree of $G$.  We discuss other extensions for $G \square H$, where $G$ is such that $V(G)$ can be partitioned into odd cycles and complete graphs, and $H$ is a graph containing a Hamiltonian path. We apply these bounds to get chromatic-choosable Cartesian products, in fact we show that these families of graphs have $\chi(G) = AT(G)$, improving previously known bounds.

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 Cartesian products of graphs as subgraphs of de Bruijn graphs of dimension at least three

1997 ◽  
Vol 79 (1-3) ◽  
pp. 3-34 ◽  
Author(s):  
Thomas Andreae ◽  
Martin Hintz ◽  
Michael Nölle ◽  
Gerald Schreiber ◽  
Gerald W. Schuster ◽  
...  
Keyword(s):  
Cartesian Products ◽  
De Bruijn Graphs ◽  
De Bruijn ◽  
Cartesian Products Of Graphs

The Maximum Genus of Cartesian Products of Graphs

1974 ◽  
Vol 26 (5) ◽  
pp. 1025-1035 ◽  
Author(s):  
Joseph Zaks
Keyword(s):  
Connected Graph ◽  
Connected Components ◽  
Open Disk ◽  
Maximum Genus ◽  
Cartesian Products ◽  
Euler’S Formula ◽  
Cartesian Products Of Graphs

The maximum genus γM(G) of a connected graph G has been defined in [2] as the maximum g for which there exists an embedding h : G —> S(g), where S(g) is a compact orientable 2-manifold of genus g, such that each one of the connected components of S(g) — h(G) is homeomorphic to an open disk; such an embedding is called cellular. If G is cellularly embedded in S(g), having V vertices, E edges and F faces, then by Euler's formulaV-E + F = 2-2g.

Blocking zero forcing processes in Cartesian products of graphs

2020 ◽  
Vol 285 ◽  
pp. 380-396
Author(s):  
Nathaniel Karst ◽  
Xierui Shen ◽  
Denise Sakai Troxell ◽  
MinhKhang Vu
Keyword(s):  
Zero Forcing ◽  
Cartesian Products ◽  
Cartesian Products Of Graphs

On the b-chromatic number of cartesian products

2018 ◽  
Vol 239 ◽  
pp. 82-93 ◽  
Author(s):  
Chuan Guo ◽  
Mike Newman
Keyword(s):  
Chromatic Number ◽  
Cartesian Products

scholarly journals Polytopality and Cartesian products of graphs

2012 ◽  
Vol 192 (1) ◽  
pp. 121-141 ◽  
Author(s):  
Julian Pfeifle ◽  
Vincent Pilaud ◽  
Francisco Santos
Keyword(s):  
Cartesian Products ◽  
Cartesian Products Of Graphs
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