11 research outputs found
Packing Coloring of Undirected and Oriented Generalized Theta Graphs
The packing chromatic number (G) of an undirected (resp.
oriented) graph G is the smallest integer k such that its set of vertices V (G)
can be partitioned into k disjoint subsets V 1,..., V k, in such a way that
every two distinct vertices in V i are at distance (resp. directed distance)
greater than i in G for every i, 1 i k. The generalized theta graph
{\ell} 1,...,{\ell}p consists in two end-vertices joined by p 2
internally vertex-disjoint paths with respective lengths 1 {\ell} 1
. . . {\ell} p. We prove that the packing chromatic number of any
undirected generalized theta graph lies between 3 and max{5, n 3 + 2}, where n
3 = |{i / 1 i p, {\ell} i = 3}|, and that both these bounds are
tight. We then characterize undirected generalized theta graphs with packing
chromatic number k for every k 3. We also prove that the packing
chromatic number of any oriented generalized theta graph lies between 2 and 5
and that both these bounds are tight.Comment: Revised version. Accepted for publication in Australas. J. Combi
Packing Coloring of Some Undirected and Oriented Coronae Graphs
The packing chromatic number χρ(G) of a graph G is the smallest integer k such that its set of vertices V(G) can be partitioned into k disjoint subsets V1, . . . , Vk, in such a way that every two distinct vertices in Vi are at distance greater than i in G for every i, 1 ≤ i ≤ k. For a given integer p ≥ 1, the p-corona of a graph G is the graph obtained from G by adding p degree-one neighbors to every vertex of G. In this paper, we determine the packing chromatic number of p-coronae of paths and cycles for every p ≥ 1
Packing coloring of some undirected and oriented coronae graphs
International audienceThe packing chromatic number \pcn(G) of a graph is the smallest integer such that its set of vertices can be partitioned into disjoint subsets , \ldots, , in such a way that every two distinct vertices in are at distance greater than in for every , .For a given integer , the generalized corona of a graph is the graph obtained from by adding degree-one neighbors to every vertex of .In this paper, we determine the packing chromatic number of generalized coronae of paths and cycles.Moreover, by considering digraphs and the (weak) directed distance between vertices, we get a natural extension of the notion of packing coloring to digraphs. We then determine the packing chromatic number of orientations of generalized coronae of paths and cycles
Packing Coloring of Some Undirected and Oriented Coronae Graphs
The packing chromatic number χρ(G) of a graph G is the smallest integer k such that its set of vertices V(G) can be partitioned into k disjoint subsets V1, . . ., Vk, in such a way that every two distinct vertices in Vi are at distance greater than i in G for every i, 1 ≤ i ≤ k. For a given integer p ≥ 1, the p-corona of a graph G is the graph obtained from G by adding p degree-one neighbors to every vertex of G. In this paper, we determine the packing chromatic number of p-coronae of paths and cycles for every p ≥ 1.
Moreover, by considering digraphs and the (weak) directed distance between vertices, we get a natural extension of the notion of packing coloring to digraphs. We then determine the packing chromatic number of orientations of p-coronae of paths and cycles