A Dirac-Type Theorem for Uniform Hypergraphs

yDirac (Proc Lond Math Soc (3) 2:69-81, 1952) proved that every connected graph of order n > 2k + 1 with minimum degree more than k contains a path of length at least 2k + 1. In this article, we give a hypergraph extension of Dirac's theorem: Given positive integers n, k and r, let H be a connected n-vertex r-graph with no Berge path of length 2k + 1. (1) If k > r >= 4 and n > 2k + 1, then delta(1)(H) <= (k r-1). Furthermore, there exist hypergraphs S-r' (n, k), Sr (n, k) and S(sK(k+1)((r)), 1) such that the equality holds if and only if S-r' (n, k) subset of H subset of S-r (n, k) or H congruent to S(sK(k+1)((r)) , 1); (2) If k >= r >= 2 and n > 2k(r - 1), then delta(1)(H) <= (k r-1). As an application of (1), we give a better lower bound of the minimum degree than the ones in the Dirac-type results for Berge Hamiltonian cycle given by Bermond et al. (Hypergraphes Hamiltoniens. In: Problemes combinatoires et theorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976). Colloq. Internat. CNRS, vol. 260, pp. 39-43. CNRS, Paris, 1976) or Clemens et al. (Electron Notes Discrete Math 54:181-186, 2016), respectively.