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Game coloring the Cartesian product of graphs
 J. Graph Theory
, 2008
"... DOI 10.1002/jgt.20338 ..."
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GAME CHROMATIC NUMBER OF CARTESIAN PRODUCT GRAPHS
"... The game chromatic number χg is considered for the Cartesian product G ✷ H of two graphs G and H. We determine exact values of χg(G✷H) when G and H belong to certain classes of graphs, and show that, in general, the game chromatic number χg(G✷H) is not bounded from above by a function of game chrom ..."
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The game chromatic number χg is considered for the Cartesian product G ✷ H of two graphs G and H. We determine exact values of χg(G✷H) when G and H belong to certain classes of graphs, and show that, in general, the game chromatic number χg(G✷H) is not bounded from above by a function of game chromatic numbers of graphs G and H. An analogous result is proved for the game coloring number colg(G✷H) of the Cartesian product of graphs.
On the acyclic chromatic number of Hamming graphs
"... Abstract. An acyclic coloring of a graph G is a proper coloring of the vertex set of G such that G contains no bichromatic cycles. The acyclic chromatic number of a graph G is the minimum number k such that G has an acyclic coloring with k colors. In this paper, acyclic colorings of Hamming graphs, ..."
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Abstract. An acyclic coloring of a graph G is a proper coloring of the vertex set of G such that G contains no bichromatic cycles. The acyclic chromatic number of a graph G is the minimum number k such that G has an acyclic coloring with k colors. In this paper, acyclic colorings of Hamming graphs, products of complete graphs, are considered. Upper and lower bounds on the acyclic chromatic number of Hamming graphs are given.
#A26 INTEGERS 13 (2013) ON MULTIPLICATIVE SIDON SETS
"... Fix integers b> a ⇧ 1 with g: = gcd(a, b). A set S ⌅ N is {a, b}multiplicative if ax ⌦ = by for all x, y S. For all n, we determine an {a, b}multiplicative set with maximum cardinality in [n], and conclude that the maximum density of an {a, b}multiplicative set is bb+g. For A,B ⌅ N, a set S ⌅ ..."
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Fix integers b> a ⇧ 1 with g: = gcd(a, b). A set S ⌅ N is {a, b}multiplicative if ax ⌦ = by for all x, y S. For all n, we determine an {a, b}multiplicative set with maximum cardinality in [n], and conclude that the maximum density of an {a, b}multiplicative set is bb+g. For A,B ⌅ N, a set S ⌅ N is {A,B}multiplicative if for all a A and b B and x, y S, the only solutions to ax = by have a = b and x = y. For 1 < a < b < c and a, b, c coprime, we give a O(1) time algorithm to approximate the maximum density of an {{a}, {b, c}}multiplicative set to arbitrary given precision. 1.
COLOURINGS OF THE CARTESIAN PRODUCT OF GRAPHS AND MULTIPLICATIVE SIDON SETS
, 2005
"... Let F be a family of connected bipartite graphs, each with at least three vertices. A proper vertex colouring of a graph G with no bichromatic subgraph in F is Ffree. The Ffree chromatic number χ(G,F) of a graph G is the minimum number of colours in an Ffree colouring of G. For appropriate choice ..."
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Let F be a family of connected bipartite graphs, each with at least three vertices. A proper vertex colouring of a graph G with no bichromatic subgraph in F is Ffree. The Ffree chromatic number χ(G,F) of a graph G is the minimum number of colours in an Ffree colouring of G. For appropriate choices of F, several wellknown types of colourings fit into this framework, including acyclic colourings, star colourings, and distance2 colourings. This paper studies Ffree colourings of the cartesian product of graphs. Let H be the cartesian product of the graphs G1,G2,...,Gd. Our main result establishes an upper bound on the Ffree chromatic number of H in terms of the maximum Ffree chromatic number of the Gi and the following numbertheoretic concept. A set S of natural numbers is kmultiplicative Sidon if ax = by implies a = b and x = y whenever x,y∈S and 1≤a,b≤k. Suppose that χ(Gi,F)≤k and S is a kmultiplicative Sidon set of cardinality d. We prove that χ(H,F) ≤ 1+2k ·maxS. We then prove that the maximum density of a kmultiplicative Sidon set is Θ(1 / logk). It follows that χ(H,F)≤O(dk logk). We illustrate the method with numerous examples, some of which generalise or improve upon existing results in the literature. 1.
Colouring the Square of the Cartesian Product of Trees
"... We prove upper and lower bounds on the chromatic number of the square of the cartesian product of trees. The bounds are equal if each tree has even maximum degree. ..."
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We prove upper and lower bounds on the chromatic number of the square of the cartesian product of trees. The bounds are equal if each tree has even maximum degree.
The 2distance coloring of the Cartesian product of cycles using optimal Lee codes
"... LetCm bethecycleof length m. WedenotetheCartesian productof ncopies of Cm by G(n,m): = Cm□Cm□···□Cm. The kdistance chromatic number χk(G) of a graph G is χ(Gk) where Gk is the kth power of the graph G = (V,E) in which two distinct vertices areadjacent inGk if andonlyif theirdistanceinGis at mostk. ..."
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LetCm bethecycleof length m. WedenotetheCartesian productof ncopies of Cm by G(n,m): = Cm□Cm□···□Cm. The kdistance chromatic number χk(G) of a graph G is χ(Gk) where Gk is the kth power of the graph G = (V,E) in which two distinct vertices areadjacent inGk if andonlyif theirdistanceinGis at mostk. Thekdistance chromatic number of G(n,m) is related to optimal codes over the ring of integers modulo m with minimum Lee distance k +1. In this paper, we consider χ2(G(n,m)) for n = 3 and m ≥ 3. In particular, we compute exact values of χ2(G(3,m)) for 3 ≤ m ≤ 8 and m = 4k, and upper bounds for m = 3k or m = 5k, for any positive integer k. We also show that the maximal size of a code in Z3 6 with minimum Lee distance 3 is 26.