### 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).

#### Keywords

#### Full Text:

PDF#### References

L. S. Chandran, A. Das, and C. D. Shah, Cubicity, boxicity, and vertex cover, Discrete Math. 309 (2009), 2488-2496.

L. S. Chandran, M. C. Francis, and R. Mathew, Chordal bipartite graphs with high boxicity, Graphs Combin. 27 (2011), 353-362.

L. S. Chandran, W. Imrich, R. Mathew, and D. Rajendraprasad, Boxicity and cubicity of product graphs, European J. Combin. 48 (2015), 100-109.

L. S. Chandran, R. Mathew, and N. Sivadasan, Boxicity of line graphs, Discrete Math. 311 (2011), 2359-2367.

L. S. Chandran and N. Sivadasan, Boxicity and treewidth, J. Combin. Theory Ser. B 97 (2007), 733-744.

M. B. Cozzens, Higher and multidimensional analogues of interval graphs, Ph.D. thesis, Rutgers University, New Brunswick, NJ, 1981.

M. B. Cozzens and F. S. Roberts, Computing the boxicity of a graph by covering its complement by cointerval graphs, Discrete Appl. Math. 6 (1983), 217-228.

M. Larsen, J. Propp, and D. Ullman, The fractional chromatic number of Mycielski's graphs, J. Graph Theory 19 (1995), 411-416.

C. G. Lekkerkerker and J. C. Boland, Representation of a finite graph by a set of intervals on the real line, Fund. Math. 51 (1962), 45-64.

W. Lin, J. Wu, P. C. B. Lam, and G. Gu, Several parameters of generalized Mycielskians, Discrete Appl. Math. 154 (2006), 1173-1182.

J. Mycielski, Sur le coloriage des graphes, Colloq. Math. 3 (1955), 161-162.

R. J. Opsut and F. S. Roberts, The Theory and Applications of Graphs, ch. On the feet maintenance, mobile radio frequency, task assignment, and traffic phasing problems, pp. 479-492, Wiley, New York, 1981.

F. S. Roberts, Recent progress in combinatorics, ch. On the boxicity and cubicity of a graph, pp. 301-310, Academic Press, New York, 1969.

F. S. Roberts, Discrete Mathematical Models, with Applications to Social, Biological, and Environmental Problems, Prentice-Hall, New Jersey, 1976.

F. S. Roberts, Theory and Applications of Graphs, Lecture Notes in Mathematics, vol. 642, ch. Food webs, competition graphs, and the boxicity of ecological phase space, pp. 447-490, Springer-Verlag, 1978.

E. R. Scheinerman, Intersection classes and multiple intersection parameters, Ph.D. thesis, Princeton University, 1984.

C. Thomassen, Interval representations of planar graphs, J. Combin. Theory Ser. B 40 (1986), 9-20.

W. T. Trotter Jr., A characterization of Roberts' inequality for boxicity, Discrete Math. 28 (1979), 303-313.

## Comments on this article

by agencia noticias (2018-03-14)

by agencia noticias (2018-04-19)

by tui la (2018-04-25)

by haytham هاوس (2018-05-05)

by Aparecida Rodrigues (2018-05-27)

by irissss nateeee (2018-06-28)

by seorado potifyaa hissense (2018-07-03)

by seorado potifyaa hissense (2018-07-08)

by Bát Tràng Gốm Sứ Quà Tặng (2018-07-14)

by seorado potifyaa hissense (2018-07-19)

by Sam Morgan (2018-08-16)

by seorado potifyaa hissense (2018-08-20)