By Peter Barlow

ISBN-10: 1429700467

ISBN-13: 9781429700467

Barlow P. An trouble-free research of the idea of numbers (Cornell college Library, 1811)(ISBN 1429700467)

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**Extra resources for An elementary investigation of the theory of numbers**

**Example text**

In particular, if δ( G ) ≥ 12 νG , then G is hamiltonian. Proof. Since dG (u) + dG (v) ≥ νG for all nonadjacent vertices, we have cl ( G ) = Kn for n = νG , and thus G is hamiltonian. The second claim is immediate, since now dG (u) + dG (v) ≥ νG for all u, v ∈ G whether adjacent or not. ⊔ ⊓ Chvátal’s condition The hamiltonian problem of graphs has attracted much attention, at least partly because the problem has practical significance. ) There are some general improvements of the previous results of this chapter, and quite many improvements in various special cases, where the graphs are somehow restricted.

Therefore M is not a maximum matching. (⇐) Assume N is a maximum matching, but M is not. Hence | N | > | M |. Consider the subgraph H = G [ M △ N ] for the symmetric difference M △ N. We have d H (v) ≤ 2 for each v ∈ H, because v is an end of at most one edge in M and N. 3, each connected component A of H is either a path or a cycle. Since no v ∈ A can be an end of two edges from N or from M, each connected component (path or a cycle) A alternates between N and M. Now, since | N | > | M |, there is a connected component A of H, which has more edges from N than from M.

Therefore χ( G t ) = k + 1. Now using inductively the above construction starting from the triangle-free graph K2 , we obtain larger triangle -free graphs with high chromatic numbers. Critical graphs D EFINITION . A k-chromatic graph G is said to be k-critical, if χ( H ) < k for all H ⊆ G with H = G. In a critical graph an elimination of any edge and of any vertex will reduce the chromatic number: χ( G −e) < χ( G ) and χ( G −v) < χ( G ) for e ∈ G and v ∈ G. Each Kn is n-critical, since in Kn −(uv) the vertices u and v can gain the same colour.

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