Degree sequences in complexes and hypergraphs

Author:
A. K. Dewdney

Journal:
Proc. Amer. Math. Soc. **53** (1975), 535-540

MSC:
Primary 05C99; Secondary 55A15

DOI:
https://doi.org/10.1090/S0002-9939-1975-0384610-1

MathSciNet review:
0384610

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Abstract: Given an -complex and a vertex in , the -*degree* of is the number of -simplexes in containing . The set of all -degrees in a complex is called its -*degree sequence* when arranged in nonincreasing order. The question ``Which sequences of integers are -degree sequences?'' is answered in this paper. This is done by generalizing the iterative characterization for the -dimensional (graphical) case due to V. Havel. A corollary to this general theorem yields the analogous generalization for -graphs. The characterization of P. Erdös and T. Gallai is discussed briefly.

**[1]**C. Berge,*Graphes et hypergraphes*, Dunod, Paris, 1970. MR**0357173 (50:9641)****[2]**P. Erdös and T. Gallai,*Graphs with prescribed degrees of vertices*, Mat. Lapok**11**(1960), 264-274.**[3]**S. Hakimi,*On the realizability of a set of integers as degrees of the vertices of a linear graph*. I, J. Soc. Indust. Appl. Math.**10**(1962), 496-506. MR**26**#5558. MR**0148049 (26:5558)****[4]**V. Havel,*Eine Bemerkung über die Existenz der endlichen Graphen*, Časopis Pěst. Mat.**80**(1955), 477-480. (Czech) MR**19**, 627. MR**0089165 (19:627c)****[5]**P. J. Hilton and S. Wylie,*Homology theory: An introduction to algebraic topology*, Cambridge Univ. Press, New York, 1960. MR**22**#5963. MR**0115161 (22:5963)**

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DOI:
https://doi.org/10.1090/S0002-9939-1975-0384610-1

Article copyright:
© Copyright 1975
American Mathematical Society