Strong chromatic index of unit distance graphs

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 inR 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.