Coloring Powers of Planar Graphs

Backbone colorings for graphs: Tree and path backbones

“…How far away from χ(G) can bbc(G, T ) be in the worst case? For each integer k ≥ 1 we define T(k) := max {bbc(G, T ) : G a graph with spanning tree T, and χ(G) = k} (1) It turns out that T(k) behaves quite primitively:…”

“…Much of the research has been concentrated on the case that G 1 is a planar graph. We refer to [1,3,4,18,21,22] for more details. In some versions of this problem one puts the additional restriction on G 1 that the colors should be sufficiently separated, in order to model practical frequency assignment problems in which interference should be kept at an acceptable level.…”

“…In some versions of this problem one puts the additional restriction on G 1 that the colors should be sufficiently separated, in order to model practical frequency assignment problems in which interference should be kept at an acceptable level. One way to model this is to use positive integers for the colors (modeling certain frequency channels) and to ask for a coloring of G 1 and G 2 such that the colors on adjacent vertices in G 2 are different, whereas they differ by at least 2 on adjacent vertices in G 1 . This problem is known as the L(2, 1)-labeling problem and has been studied (under various names) in [2,[7][8][9][10][11]19].…”

Coloring the square of a planar graph

“…Better bounds were then obtained for large values of ∆. It was shown that χ(G 2 ) ≤ [1], and that χ(G 2 ) ≤ 9 5 ∆ + 1 for ∆ ≥ 47 by Borodin et al [2]. Finally, the best known upper bound so far has been obtained by Molloy and Salavatipour [12] …”

On the connectivity of k-distance graphs

“…Let G be a graph and k be a positive integer. The k−power of G, denoted by G k , is the graph with the vertex set V (G k ) = V (G) and the edge set E(G k ) = {xy|1 ≤ d(x, y) ≤ k} where d(x, y) is the distance between two vertices of x, y ∈ V (G) on the shortest path between them [1]. The k−subdivision of a graph G, denoted by G 1 k , is constructed by replacing each edge xy of G with path of length k. The vertices of G in G 1 k are as the main vertices of G. The union of graphs G 1 and G 2 with disjoint point sets V 1 and V 2 and edge sets E 1 and E 2 is the graph G = G 1 ∪ G 2 with V = V 1 ∪ V 2 and E = E 1 ∪ E 2 [5].…”