On multigraphic and potentially multigraphic sequences

Dedicated to the memory of Antal Iványi

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An r-graph(or a multigraph) is a loopless graph in which no two vertices are joined by more than r edges. An r-complete graph on n vertices, denoted by Kn(r) , is an r-graph on n vertices in which each pair of vertices is joined by exactly r edges. A non-increasing sequence π = (d1, d2, ..., dn) of non-negative integers is said to be r-graphic if it is realizable by an r-graph on n vertices. An r-graphic sequence π is said to be potentially SL;M(r) -graphic if it has a realization containing SL;M(r) as a subgraph. We obtain conditions for an r-graphic sequence to be potentially S(r) L;M-graphic. These are generalizations from split graphs to p-tuple r-split graph.

[1] D. de Caen, An upper bound on the sum of squares of degrees in a graph, Discrete Math. 185 (1998) 245–248. )38

[2] V. Chungphaisan, Conditions for a sequences to be r-graphic, Discrete Math. 7 (1974) 31–39. )36, 37

[3] P. Erdős, T. Gallai, Graphs with prescribed degrees (in Hungarian) Matematikai Lapok 11(1960) 264–274. )36

[4] P. Erdős, M. S. Jacobson, J. Lehel, Graphs realizing the same degree sequences and their respective clique numbers, in: Graph Theory, Combinatorics and Ap- plications, vol. 1, John Wiley and Sons, New York, 1991, 439–449. )37

[5] D. R. Fulkerson, A. J. Ho_man, M. H. McAndrew, Some properties of graphs with multiple edges, Canad. J. Math. 17 (1965) 166–177. )36

[6] D. J. Kleitman, D. L. Wang, Algorithm for constructing graphs and digraphs with given valencies and factors, Discrete Math. 6 (1973) 79–88. )36

[7] J. S. Li, Z. X. Song, R. Luo, The Erdős-Jacobson-Lehel conjecture on potentially pk-graphic sequences is true, Sci. China Ser. A 41 (1998) 510–520. )37

[8] J. S. Li, Z. X. Song, An extremal problem on the potentially pk-graphic sequence, Discrete Math. 212 (2000) 223–231. )37

[9] S. Pirzada, An Introduction to Graph Theory, Universities Press, Orient Blak- swan, Hyderabad, India 2012. )36

[10] S. Pirzada, B. A. Chat, Potentially graphic sequences of split graphs, Kragujevac J. Math. 38, 1 (2014) 73–81. )37, 38

[11] S. Pirzada, B. A. Chat, F. A. Dar, Graphical sequences of some family of inducedsubgraphs, J. Algebra Comb. Discrete Appl. 2(2) (2015) 95–109. )38

[12] A. R. Rao, The clique number of a graph with a given degree sequence, Proc. Symposium on Graph Theory (ed. A. R. Rao, Macmillan and Co. India Ltd, I.S.I. Lecture Notes Series, 4 (1979) 251–267. )37

[13] J. H. Yin, Conditions for r-graphic sequences to be potentially Kr m+1 -graphic, Discrete Math. 309 (2009) 6271–6276. )37

[14] J. H. Yin, A Havel-Hakimi type procedure and a sufficient condition for a se- quence to be potentially Sr;s-graphic, Czechoslovak Math. J. 62,3 (2012) 863– 867. )37, 38

Acta Universitatis Sapientiae, Informatica

The Journal of "Sapientia" Hungarian University of Transylvania

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