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The smallest degree sum that yields potentially P k ‐graphical sequences
Author(s) -
JiongSheng Li,
ZiXia Song
Publication year - 1998
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/(sici)1097-0118(199810)29:2<63::aid-jgt2>3.0.co;2-a
Subject(s) - mathematics , combinatorics , degree (music) , conjecture , simple graph , discrete mathematics , graph , physics , acoustics
A simple graph G is said to have property P k if it contains a complete subgraph of order k + 1, and a sequence π is potentially P k ‐graphical if it has a realization having property P k . Let σ ( k , n ) denote the smallest degree sum such that every n ‐term graphical sequence π without zero terms and with degree sum σ(π) ≥ σ( k , n ) is potentially P k ‐graphical. Erdós, Jacobson, and Lehel [Graph Theory, 1991, 439–449] conjectured that σ( k , n ) = ( k − 1)(2 n − k ) + 2. In this article, we prove that the conjecture is true for k = 4 and n ≥ 10. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 63–72, 1998