Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
with one new vertex adjacent to two clique vertices satisfies Ore's condition but not Dirac's condition
Example
Start with the clique on and add a vertex adjacent exactly to and . The resulting five-vertex graph satisfies Ore's condition but not Dirac's minimum-degree condition, and it has the Hamilton cycle .
Facts & Assumptions
Given: The explicitly described five-vertex graph .
A complete graph contains every edge between its distinct vertices (Empty and complete graphs, complete bipartite graphs, and the convention that and have vertices).
Degree counts neighbours, and minimum degree is the least vertex degree (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).
Ore's condition implies Hamiltonicity for graphs of order at least three (Ore's theorem: an -vertex graph with and for every nonadjacent pair is Hamiltonian).
Dirac's sufficient condition is (Dirac's theorem: every -vertex graph with and is Hamiltonian).
Verification
The degree multiset is : has degree two, have degree four, and have degree three. Thus , so Dirac's hypothesis [L2] fails.
Directly, every consecutive pair in is an edge, and the list visits all five vertices once before returning.
The only nonedges are and . Each has endpoint-degree sum , so Ore's condition holds; [L1] therefore confirms Hamiltonicity.
Hence Ore's condition can hold when Dirac's condition does not, and the displayed Hamilton cycle verifies the conclusion.
Depends on
- Ore's theorem: an $n$-vertex graph with $n\ge3$ and $\deg(u)+\deg(v)\ge n$ for every nonadjacent pair is Hamiltonian
- Dirac's theorem: every $n$-vertex graph with $n\ge3$ and $\delta(G)\ge n/2$ is Hamiltonian
- Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree
- Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices
Used by
Nothing in the library uses this result yet.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 22 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Applied Combinatorics, Eulerian and Hamiltonian Graphs (standard reference, not scraped)