Chromatic number of prime distance graphs
2nd Twente workshop on Graphs and combinatorial optimization
On the packing chromatic number of Cartesian products, hexagonal lattice, and trees
Discrete Applied Mathematics
Fractional chromatic number of distance graphs generated by two-interval sets
European Journal of Combinatorics
The packing chromatic number of infinite product graphs
European Journal of Combinatorics
Complexity of the packing coloring problem for trees
Discrete Applied Mathematics
On the packing chromatic number of some lattices
Discrete Applied Mathematics
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The packing chromatic number@g"@r(G) of a graph G is the least integer k for which there exists a mapping f from V(G) to {1,2,...,k} such that any two vertices of color i are at a distance of at least i+1. This paper studies the packing chromatic number of infinite distance graphs G(Z,D), i.e. graphs with the set Z of integers as vertex set, with two distinct vertices i,j@?Z being adjacent if and only if |i-j|@?D. We present lower and upper bounds for @g"@r(G(Z,D)), showing that for finite D, the packing chromatic number is finite. Our main result concerns distance graphs with D={1,t} for which we prove some upper bounds on their packing chromatic numbers, the smaller ones being for t=447: @g"@r(G(Z,{1,t}))@?40 if t is odd and @g"@r(G(Z,{1,t}))@?81 if t is even.