Ore's theorem

Ore's theorem is a result in graph theory proved in 1960 by Norwegian mathematician Øystein Ore. It gives a sufficient condition for a graph to be Hamiltonian, essentially stating that a graph with "sufficiently many edges" must contain a Hamilton cycle. Specifically, the theorem considers the sum of the degrees of pairs of non-adjacent vertices: if every such pair has a sum that at least equals the total number of vertices in the graph, then the graph is Hamiltonian.

Formal statement

Let G be a (finite and simple) graph with n ≥ 3 vertices. We denote by deg v the degree of a vertex v in G, i.e. the number of incident edges in G to v. Then, Ore's theorem states that if

deg v + deg w ≥ n for every pair of distinct non-adjacent vertices v and w of G (*)

then G is Hamiltonian.

Proof

It is equivalent to show that every non-Hamiltonian graph G does not obey condition (*). Accordingly, let G be a graph on n ≥ 3 vertices that is not Hamiltonian, and let H be formed from G by adding edges one at a time that do not create a Hamiltonian cycle, until no more edges can be added. Let x and y be any two non-adjacent vertices in H. Then adding edge xy to H would create at least one new Hamiltonian cycle, and the edges other than xy in such a cycle must form a Hamiltonian path v₁v₂...v_n in H with x = v₁ and y = v_n. For each index i in the range 2 ≤ i ≤ n, consider the two possible edges in H from v₁ to v_i and from v_{i − 1} to v_n. At most one of these two edges can be present in H, for otherwise the cycle v₁v₂...v_{i − 1}v_nv_{n − 1}...v_i would be a Hamiltonian cycle. Thus, the total number of edges incident to either v₁ or v_n is at most equal to the number of choices of i, which is n − 1. Therefore, H does not obey property (*), which requires that this total number of edges (deg v₁ + deg v_n) be greater than or equal to n. Since the vertex degrees in G are at most equal to the degrees in H, it follows that G also does not obey property (*).

Algorithm

Palmer (1997) describes the following simple algorithm for constructing a Hamiltonian cycle in a graph meeting Ore's condition.

1. Arrange the vertices arbitrarily into a cycle, ignoring adjacencies in the graph.
2. While the cycle contains two consecutive vertices v_i and v_i + 1 that are not adjacent in the graph, perform the following two steps:
   • Search for an index j such that the four vertices v_i, v_i + 1, v_j, and v_j + 1 are all distinct and such that the graph contains edges from v_i to v_j and from v_j + 1 to v_i + 1
   • Reverse the part of the cycle between v_i + 1 and v_j (inclusive).

Each step increases the number of consecutive pairs in the cycle that are adjacent in the graph, by one or two pairs (depending on whether v_j and v_j + 1 are already adjacent), so the outer loop can only happen at most n times before the algorithm terminates, where n is the number of vertices in the given graph. By an argument similar to the one in the proof of the theorem, the desired index j must exist, or else the nonadjacent vertices v_i and v_i + 1 would have too small a total degree. Finding i and j, and reversing part of the cycle, can all be accomplished in time O(n). Therefore, the total time for the algorithm is O(n²), matching the number of edges in the input graph.

Related results

Ore's theorem is a generalization of Dirac's theorem that, when each vertex has degree at least n/2, the graph is Hamiltonian. For, if a graph meets Dirac's condition, then clearly each pair of vertices has degrees adding to at least n.

In turn Ore's theorem is generalized by the Bondy–Chvátal theorem. One may define a closure operation on a graph in which, whenever two nonadjacent vertices have degrees adding to at least n, one adds an edge connecting them; if a graph meets the conditions of Ore's theorem, its closure is a complete graph. The Bondy–Chvátal theorem states that a graph is Hamiltonian if and only if its closure is Hamiltonian; since the complete graph is Hamiltonian, Ore's theorem is an immediate consequence.

Woodall (1972) found a version of Ore's theorem that applies to directed graphs. Suppose a digraph G has the property that, for every two vertices u and v, either there is an edge from u to v or the outdegree of u plus the indegree of v equals or exceeds the number of vertices in G. Then, according to Woodall's theorem, G contains a directed Hamiltonian cycle. Ore's theorem may be obtained from Woodall by replacing every edge in a given undirected graph by a pair of directed edges. A closely related theorem by Meyniel (1973) states that an n-vertex strongly connected digraph with the property that, for every two nonadjacent vertices u and v, the total number of edges incident to u or v is at least 2n − 1 must be Hamiltonian.

Ore's theorem may also be strengthened to give a stronger condition than Hamiltonicity as a consequence of the degree condition in the theorem. Specifically, every graph satisfying the conditions of Ore's theorem is either a regular complete bipartite graph or is pancyclic (Bondy 1971).

References

• Bondy, J. A. (1971), "Pancyclic graphs I", Journal of Combinatorial Theory, Series B, 11 (1): 80–84, doi:10.1016/0095-8956(71)90016-5 .
• Meyniel, M. (1973), "Une condition suffisante d'existence d'un circuit hamiltonien dans un graphe orienté", Journal of Combinatorial Theory, Series B (in French), 14 (2): 137–147, doi:10.1016/0095-8956(73)90057-9 .
• Ore, Ø. (1960), "Note on Hamilton circuits", American Mathematical Monthly, 67 (1): 55, doi:10.2307/2308928, JSTOR 2308928 .
• Palmer, E. M. (1997), "The hidden algorithm of Ore's theorem on Hamiltonian cycles", Computers & Mathematics with Applications, 34 (11): 113–119, doi:10.1016/S0898-1221(97)00225-3, MR 1486890 .
• Woodall, D. R. (1972), "Sufficient conditions for circuits in graphs", Proceedings of the London Mathematical Society, Third Series, 24: 739–755, doi:10.1112/plms/s3-24.4.739, MR 0318000 .