A note onS-packing colorings of lattices

Abstract

a b s t r a c tLet a 1 , a 2 , . . . , a k be positive integers. An (a 1 , a 2 , . . . , a k )-packing coloring of a graph G is a mapping from V (G) to {1, 2, . . . , k} such that vertices with color i have pairwise distance greater than a i . In this paper, we study (a 1 , a 2 , . . . , a k )-packing colorings of several lattices including the infinite square, triangular, and hexagonal lattices. For k small, we determine all a i such that these graphs have packing colorings. We also give some exact values and asymptotic bounds.

On S-packing edge-colorings of cubic graphs

“…The vertex analogous of S-packing edge-coloring has been first studied by Goddard and Xu [16,17] and then recently on cubic graphs [2,3,4,12,14]. The particular case of (1, 2, .…”

Packing Chromatic Number of Subdivisions of Cubic Graphs

“…If 1 a / ∈ f 2 (N (w 4 ) − w 1 ), then we obtain a feasible coloring on G by switching colors of w 1 and w 4 , assigning 1 a to w, and using f 2 on other vertices. Therefore, by (18), we may assume f 2 (N (w 4 ) − w 1 ) = {1 a , 1 b }. Similarly, by (18), we may assume f 2 (N (w 6 ) − w 2 ) = {1 a , 1 b } (See Figure 3).…”

“…Therefore, by (18), we may assume f 2 (N (w 4 ) − w 1 ) = {1 a , 1 b }. Similarly, by (18), we may assume f 2 (N (w 6 ) − w 2 ) = {1 a , 1 b } (See Figure 3). With (14), (17), and the case, 3 a / ∈ f 2 (B(w, 3) − {w}) and we can extend f 2 to G by assigning f 2 (w) = 3 a .…”

Subdivision into i-packings and S-packing chromatic number of some lattices

“…, k) first appear in [7,3]. Goddard and Xu [8] have recently studied S-packing colorings for the infinite path and for square and triangular lattices, determining conditions on the first elements of the sequence for which the graph is or is not S-packingcolorable.…”

“…However, determining the families of graphs G for which for any S such that G is S-colorable, the S-coloring satisfies the above property is satisfied remains an open question. Recently Goddard and Xu [8] proved that there exist nondecreasing sequences S such that P ∞ is S-colorable and in any (s 1 , . .…”