 TOPICS  # Ore Graph

An Ore graph is a graph that satisfies Ore's theorem, i.e., a graph for which the sums of the degrees of nonadjacent vertices is greater than or equal to the number of nodes for all subsets of nonadjacent vertices. Let be the vertex set and the edge set of , let denote the vertex count of , let denote the vertex degree of vertex by , and let denote a pair of vertices. The condition for to be an Ore graph can then be written (Ore 1960; Bondy 1971). Note that Skiena (1990, p. 197) and Pemmaraju and Skiena (2003, p. 301) state a weaker version of this result with the "greater or equal than" replaced by "greater than." Ore's theorem states that Ore graphs are always Hamiltonian. Furthermore, for such graphs, a Hamiltonian cycle can be constructed in polynomial time (Bondy and Chvátal 1976; Skiena 1990, p. 197). The numbers of graphs on , 2, ... satisfying Ore's criterion (using the "vacuous truth" convention and so including complete graphs ) are 1, 1, 1, 3, 5, 21, 68, 503, 4942, 128361, ... (OEIS A264683), the first few of which are illustrated above. Every graph satisfying Ore's criterion is Hamiltonian, but not every Hamiltonian graph satisfies the criterion. The numbers of simple Hamiltonian graphs on , 2, ... vertices that do not satisfy Ore's criterion are 0, 0, 0, 0, 3, 27, 315, 5693, 172141, 9176757, ... (OEIS A264684), the first few of which are illustrated above.

Hamiltonian Cycle, Hamiltonian Graph, Ore's Theorem

## Explore with Wolfram|Alpha ## References

Bondy, J. A. "Pancyclic Graphs I." J. Combin. Th. 11, 80-84, 1971.Bondy, J. A. and Chvátal, V. "A Method in Graph Theory." Disc. Math. 15, 111-136, 1976.Meyniel, M. "Une condition suffisante d'existence d'un circuit hamiltonien dans un graphe orienté." J. Combin. Th. 14, 137-147, 1973.Ore, O. "A Note on Hamiltonian Circuits." Amer. Math. Monthly 67, 55, 1960.Palmer, E. M. "The Hidden Algorithm of Ore's Theorem on Hamiltonian Cycles." Computers Math. Appl. 34, 113-119, 1997.Pemmaraju, S. and Skiena, S. Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Cambridge, England: Cambridge University Press, 2003.Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990.Sloane, N. J. A. Sequences A264683 and A264684 in "The On-Line Encyclopedia of Integer Sequences."Woodall, D. R. "Sufficient Conditions for Circuits in Graphs." Proc. London Math. Soc. 24: 739-755, 1972.

Ore Graph

## Cite this as:

Weisstein, Eric W. "Ore Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OreGraph.html