On maximal paths and circuits of graphs

TL;DR: It is proved that if G is a connected graph such with the independence number at most ⌈ n 2 ⌉ and the degree sum of any pair of non-adjacent vertices is at least n − 3 , then G is on-line arbitrarily partitionable except for two graphs of small orders.

TL;DR: In this article , the authors obtained a stability version of Dirac's result by characterizing those graphs with minimum degree $k$ and circumference at most 2k+1, where k is the number of cycles in a graph.

TL;DR: Experimental results show that the obtained model can effectively extract dependencies among features, and hence, its performance as a classifier is superior compared to other three methods.

TL;DR: In this article, the strong clique number of graphs missing some set of cycle lengths was studied and it was shown that for a graph G of large enough maximum degree Δ, ω 2 ε (G ) ≤ 5 Δ 2 ∕ 4 if G is triangle-free; ω ε(G) ≤ 3 Δ − 1 √ Δ − ε 2 for some k ≥ 2.

TL;DR: The first result in this direction was due to Turân as discussed by the authors, who proved that a graph with kn vertices and Ck, 2n+1 edges always contains a complete graph of order k + 1.