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For a 2-connected graph G on n vertices and two vertices x , y ∈ V (G) , we prove that there is an (x , y) -path of length at least k , if there are at least n − 1 2 vertices in V (G) \ { x , y } of degree at least k. This strengthens a celebrated theorem due to Erdős and Gallai in 1959. As the first application of this result, we show that a 2-connected graph with n vertices contains a cycle of length at least 2 k , if it has at least n 2 + k vertices of degree at least k. This confirms a 1975 conjecture made by Woodall. As other applications, we obtain some results which generalize previous theorems of Dirac, Erdős-Gallai, Bondy, and Fujisawa et al., present short proofs of the path case of Loebl-Komlós-Sós Conjecture which was verified by Bazgan et al. and a conjecture of Bondy on longest cycles (for large graphs) which was confirmed by Fraisse and Fournier, and make progress on a conjecture of Bermond. [ABSTRACT FROM AUTHOR]

Abstract: Let be an edge-colored graph. The color degree of a vertex of , is defined as the number of colors of the edges incident to . The color number of is defined as the number of colors of the edges in . A rainbow triangle is one in which every pair of edges have distinct colors. In this paper we give some sufficient conditions for the existence of rainbow triangles in edge-colored graphs in terms of color degree, color number and edge number. As a corollary, a conjecture proposed by Li and Wang [H. Li and G. Wang, Color degree and heterochromatic cycles in edge-colored graphs, European J. Combin. 33 (2012) 1958–1964] is confirmed. [Copyright &y& Elsevier]