Packing chromatic number versus chromatic and clique number
نویسندگان
چکیده
The packing chromatic number χρ(G) of a graphG is the smallest integer k such that the vertex set of G can be partitioned into sets Vi, i ∈ [k], where each Vi is an i-packing. In this paper, we investigate for a given triple (a, b, c) of positive integers whether there exists a graph G such that ω(G) = a, χ(G) = b, and χρ(G) = c. If so, we say that (a, b, c) is realizable. It is proved that b = c ≥ 3 implies a = b, and that triples (2, k, k + 1) and (2, k, k + 2) are not realizable as soon as k ≥ 4. Some of the obtained results are deduced from the bounds proved on the packing chromatic number of the Mycielskian. Moreover, a formula for the independence number of the Mycielskian is given. A lower bound on χρ(G) in terms of ∆(G) and α(G) is also proved.
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