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Definition: G has radius at most r if there is a vertex v such that every vertex in G can be reached from v in at most r steps. The radius of G is the smallest possible radius of any vertex, known as the center. Example: Radius of Pn is 4 with center being a middle vertex. Lemma: diameter < 2*radius Pf: any two vertices are reachable from each other by walking through the center. n(n-1) Lemma: The order of Kn is (") 2 Pf: each vertex is incident on n-1 edges, so n(n– 1) incidences overall. Since each edge has two vertices, we need to divide by two. Definition: the degree of a vertex v, degc(v) is the number of edges it is incident on. Corollary If every vertex has degree at least |V(G)|/2, then the graph is Hamiltonian. Proof: In that case deg(u) + deg(v) 2÷+ n = n for any two vertices, so Ore's 2 theorem applies.

Definition: G has radius at most r if there is a vertex v such that every vertex in G can be reached from v in at most r steps. The radius of G is the smallest possible radius of any vertex, known as the center. Example: Radius of Pn is 4 with center being a middle vertex. Lemma: diameter < 2*radius Pf: any two vertices are reachable from each other by walking through the center. n(n-1) Lemma: The order of Kn is (") 2 Pf: each vertex is incident on n-1 edges, so n(n– 1) incidences overall. Since each edge has two vertices, we need to divide by two. Definition: the degree of a vertex v, degc(v) is the number of edges it is incident on. Corollary If every vertex has degree at least |V(G)|/2, then the graph is Hamiltonian. Proof: In that case deg(u) + deg(v) 2÷+ n = n for any two vertices, so Ore's 2 theorem applies.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Definition: G has radius at most r if there is a vertex v such that every vertex in G can be reached from v in at most r steps. The radius of G is the smallest possible radius of any vertex, known as the center. Example: Radius of P is 4 with center being a middle vertex. Lemma: diameter < 2*radius Pf: any two vertices are reachable from each other by walking through the center. п(п-1) Lemma: The order of Kn is (“) %3D 2 Pf: each vertex is incident on n-1 edges, so n(n– 1) incidences overall. Since each edge has two vertices, we need to divide by two. Definition: the degree of a vertex v, degc(v) is the number of edges it is incident on. Corollary If every vertex has degree at least |V(G)|/2, then the graph is Hamiltonian. п Proof: In that case deg(u) + deg(v) >÷+ п = n for any two vertices, so Ore's theorem applies.
Transcribed Image Text:Definition: G has radius at most r if there is a vertex v such that every vertex in G can be reached from v in at most r steps. The radius of G is the smallest possible radius of any vertex, known as the center. Example: Radius of P is 4 with center being a middle vertex. Lemma: diameter < 2*radius Pf: any two vertices are reachable from each other by walking through the center. п(п-1) Lemma: The order of Kn is (“) %3D 2 Pf: each vertex is incident on n-1 edges, so n(n– 1) incidences overall. Since each edge has two vertices, we need to divide by two. Definition: the degree of a vertex v, degc(v) is the number of edges it is incident on. Corollary If every vertex has degree at least |V(G)|/2, then the graph is Hamiltonian. п Proof: In that case deg(u) + deg(v) >÷+ п = n for any two vertices, so Ore's theorem applies.
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