PDF

Discussiones Mathematicae Graph Theory 26(2) (2006) 317-333
DOI: https://doi.org/10.7151/dmgt.1323

EXTREMUM DEGREE SETS OF IRREGULAR ORIENTED GRAPHS AND PSEUDODIGRAPHS

Zyta Dziechcińska-Halamoda, Zofia Majcher and Jerzy Michael Institute of Mathematics and Informatics University of Opole Oleska 48, 45-052 Opole, Poland e-mail:  zdziech@uni.opole.pl    majcher@math.uni.opole.pl    michael@uni.opole.pl

Zdzisław Skupień Faculty of Applied Mathematics AGH University of Science and Technology al. Mickiewicza 30, 30-059 Kraków, Poland e-mail: skupien@uci.agh.edu.pl

Abstract

A digraph in which any two vertices have distinct degree pairs is called irregular. Sets of degree pairs for all irregular oriented graphs (also loopless digraphs and pseudodigraphs) with minimum and maximum size are determined. Moreover, a method of constructing corresponding irregular realizations of those sets is given.

Keywords: irregular digraphs, degree sequences, degree sets.

2000 Mathematics Subject Classification: 05C07.

References

[1]	Y. Alavi, G. Chartrand, F.R.K. Chung, P. Erdös, R.L. Graham and O.R. Oel lermann, Highly irregular graphs, J. Graph Theory 11 (1987) 235-249, doi: 10.1002/jgt.3190110214.
[2]	Y. Alavi, J. Liu and J. Wang, Highly irregular digraphs, Discrete Math. 111 (1993) 3-10, doi: 10.1016/0012-365X(93)90134-F.
[3]	G. Chartrand and L. Lesniak, Graphs and Digraphs (Chapman and Hall, Third edition, 1996).
[4]	Z. Dziechcińska-Halamoda, Z. Majcher, J. Michael and Z. Skupień, Large minimal irregular digraphs, Opuscula Mathematica 23 (2003) 21-24.
[5]	M. Gargano, J.W. Kennedy and L.V. Quintas, Irregular digraphs, Congress. Numer. 72 (1990) 223-231.
[6]	J. Górska and Z. Skupień, Near-optimal irregulation of digraphs, submitted.
[7]	J. Górska, Z. Skupień, Z. Majcher and J. Michael, A smallest irregular oriented graph containing a given diregular one, Discrete Math. 286 (2004) 79-88, doi: 10.1016/j.disc.2003.11.049.
[8]	J.S. Li and K. Yang, Degree sequences of oriented graphs, J. Math. Study 35 (2002) 140-146.
[9]	Z. Majcher and J. Michael, Degree sequences of highly irregular graphs, Discrete Math. 164 (1997) 225-236, doi: 10.1016/S0012-365X(97)84782-6.
[10]	Z. Majcher and J. Michael, Highly irregular graphs with extreme numbers of edges, Discrete Math. 164 (1997) 237-242, doi: 10.1016/S0012-365X(96)00056-8.
[11]	Z. Majcher and J. Michael, Degree sequences of digraphs with highly irregular property, Discuss. Math. Graph Theory 18 (1998) 49-61, doi: 10.7151/dmgt.1062.
[12]	Z. Majcher, J. Michael, J. Górska and Z. Skupień, The minimum size of fully irregular oriented graphs, Discrete Math. 236 (2001) 263-272, doi: 10.1016/S0012-365X(00)00446-5.
[13]	A. Selvam, Highly irregular bipartite graphs, Indian J. Pure Appl. Math. 27 (1996) 527-536.
[14]	Z. Skupień, Problems on fully irregular digraphs, in: Z. Skupień, R. Kalinowski, guest eds., Discuss. Math. Graph Theory 19 (1999) 253-255, doi: 10.7151/dmgt.1102.

Received 12 March 2005
Revised 21 October 2005