Strong chromatic index of unit distance graphs

Abstract

The strong chromatic index of a graph G, denoted by s ′ ( G ), is defined as the least number of colors in a coloring of edges of G, such that each color class is an induced matching (or: if edges e and f have the same color, then both vertices of e are not adjacent to any vertex of f). A graph G is a unit distance graph in R n if vertices of G can be uniquely identified with points in R n, so that u v is an edge of G if and only if the Euclidean distance between the points identified with u and v is 1. We would like to find the largest possible value of s ′ ( G ), where G is a unit distance graph (in R 2 and R 3) of maximum degree Δ. We show that s ′ ( G ) ≤ K Δ 2 ln Δ, where G is a unit distance graph in R 3 of maximum degree Δ. We also show that the maximum possible size of a strong clique in unit distance graph in R 3 is linear in Δ and give a tighter result 6 Δ − 3 for unit distance graphs in the plane.

Strong Cliques in Claw-Free Graphs

“…This problem has also been studied for restricted classes of graphs, including: bipartite graphs [15,16], C 4 -free graphs [25], random graphs [17], graphs of bounded degeneracy [4,7,11], planar graphs [16,22,24], unit distance graphs [10] and graphs of maximum degree 3 and 4 [1,9,[19][20][21]. In particular, the strong chromatic index of a claw-free graph with maximum degree D is at most 1:125D 2 þ D [12].…”