5 research outputs found
Polynomial algorithms that prove an NP-hard hypothesis implies an NP-hard conclusion
A number of results in Hamiltonian graph theory are of the form implies , where is a property of graphs that is NP-hard and is a cycle structure property of graphs that is also NP-hard. Such a theorem is the well-known Chv\'{a}tal-Erd\"{o}s Theorem, which states that every graph with is Hamiltonian. Here is the vertex connectivity of and is the cardinality of a largest set of independent vertices of . In another paper Chv\'{a}tal points out that the proof of this result is in fact a polynomial time construction that either produces a Hamilton cycle or a set of more than independent vertices. In this note we point out that other theorems in Hamiltonian graph theory have a similar character. In particular, we present a constructive proof of the well-known theorem of Jung for graphs on or more vertices.. \u
A Survey of Best Monotone Degree Conditions for Graph Properties
We survey sufficient degree conditions, for a variety of graph properties,
that are best possible in the same sense that Chvatal's well-known degree
condition for hamiltonicity is best possible.Comment: 25 page