### Bounds for the boxicity of Mycielski graphs

#### Abstract

A box in Euclidean k-space is the Cartesian product of k closed intervals on the real line. The boxicity of a graph G, denoted by box(G), is the minimum nonnegative integer k such that G can be isomorphic to the intersection graph of a family of boxes in Euclidean k-space.

Mycielski introduced an interesting graph operation that extends a graph G to a new graph M(G), called the Mycielski graph of G. In this paper we observe the behavior of boxicity of Mycielski graphs. We see that box(M(G)) is at least box(G) for a graph G, and hence we are interested in whether the boxicity of Mycielski graph of G is more than that of G or not. Here we give bounds for the boxicity of Mycielski graphs in terms of the number of universal vertices of G and the edge clique cover number of the complement of G. Further observations determine the boxicity of the Mycielski graph M(G) if G has no universal vertices or odd universal vertices and box(G) is equal to the edge clique cover number of the complement of G.

We also present relations between the Mycielski graph M(G) and its generalizations M3(G) and Mr(G) in the context of boxicity, which will encourage us to calculate the boxicity of M(G) and M3(G).

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