Two sufficient conditions for a graphic sequence to have a realization with prescribed clique size

A graphic sequence π = ( d 1 , d 2 , … , d n ) is said to be potentially K r + 1 -graphic, if π has a realization G containing K r + 1 , a clique of r + 1 vertices, as a subgraph. In this paper, we give two simple sufficient conditions for a graphic sequence π = ( d 1 , d 2 , … , d n ) to be potenti...

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Vydáno v:	Discrete mathematics Ročník 301; číslo 2; s. 218 - 227
Hlavní autoři:	Yin, Jian-Hua, Li, Jiong-Sheng
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Amsterdam Elsevier B.V 06.10.2005 Elsevier
Témata:	Algorithmics. Computability. Computer arithmetics Applied sciences Combinatorics Combinatorics. Ordered structures Computer science; control theory; systems Degree sequence Exact sciences and technology Graph Graph theory Mathematics Potentially [formula omitted]-graphic sequence Sciences and techniques of general use Theoretical computing Potentially K r + 1 -graphic sequence Graph Degree sequence Graphics Algorithm theory Degree sequence: Potentially Kr+1 -graphic sequence Edge(graph) Sufficient condition Graph theory Graphic sequence Graph algorithm
ISSN:	0012-365X, 1872-681X
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Shrnutí:	A graphic sequence π = ( d 1 , d 2 , … , d n ) is said to be potentially K r + 1 -graphic, if π has a realization G containing K r + 1 , a clique of r + 1 vertices, as a subgraph. In this paper, we give two simple sufficient conditions for a graphic sequence π = ( d 1 , d 2 , … , d n ) to be potentially K r + 1 -graphic. We also show that the two sufficient conditions imply a theorem due to Rao [An Erdös-Gallai type result on the clique number of a realization of a degree sequence unpublished.], a theorem due to Li et al [The Erdös–Jacobson–Lehel conjecture on potentially P k -graphic sequences is true, Sci. China Ser. A 41 (1998) 510–520.], the Erdös–Jacobson–Lehel conjecture on σ ( K r + 1 , n ) which was confirmed (see [Potentially G -graphical degree sequences, in: Y. Alavi et al. (Eds.), Combinatorics, Graph Theory, and Algorithms, vol. 1, New Issues Press, Kalamazoo Michigan, 1999, pp. 451–460; The smallest degree sum that yields potentially P k -graphic sequences, J. Graph Theory 29 (1998) 63–72; An extremal problem on the potentially P k -graphic sequence, Discrete Math. 212 (2000) 223–231; The Erdös–Jacobson–Lehel conjecture on potentially P k -graphic sequences is true, Sci. China Ser. A 41 (1998) 510–520.]) and the Yin–Li–Mao conjecture on σ ( K r + 1 - e , n ) [An extremal problem on the potentially K r + 1 - e -graphic sequences, Ars Combin. 74 (2005) 151–159.], where K r + 1 - e is a graph obtained by deleting one edge from K r + 1 .
ISSN:	0012-365X 1872-681X
DOI:	10.1016/j.disc.2005.03.028