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A graph H is called a subgraph of a graph G if V (H) ≤ V(G) and E(H) ≤ E(G); in this case we write HCG. Prove that every graph G with |E(G)| > 1 edges contains a subgraph H with minimum degree 8(H) > d(G)/2. (Hint: Construct a sequence of subgraphs G = Go 2 G₁2 as follows: if G₂ has a vertex v; of degree dc; (vi) ≤ d(G₁)/2, then obtain G₁+1 from Gį by deleting v; (which includes deleting all edges containing v;); if not, then terminate with H = G₁. Show that the resulting graph H has the desired properties.)

A graph H is called a subgraph of a graph G if V (H) ≤ V(G) and E(H) ≤ E(G); in this case we write HCG. Prove that every graph G with |E(G)| > 1 edges contains a subgraph H with minimum degree 8(H) > d(G)/2. (Hint: Construct a sequence of subgraphs G = Go 2 G₁2 as follows: if G₂ has a vertex v; of degree dc; (vi) ≤ d(G₁)/2, then obtain G₁+1 from Gį by deleting v; (which includes deleting all edges containing v;); if not, then terminate with H = G₁. Show that the resulting graph H has the desired properties.)

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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A graph H is called a subgraph of a graph G if V (H) ≤ V (G) and E(H) ≤ E(G); in this case we write HCG. Prove that every graph G with |E(G)| > 1 edges contains a subgraph H with minimum degree 8(H) > d(G)/2. (Hint: Construct a sequence of subgraphs G = Go 2 G₁ 2. as follows: if G; has a vertex vį, of degree da; (vi) ≤ d(Gi)/2, then obtain G₁+1 from Gį by deleting vį (which includes deleting all edges containing vį); if not, then terminate with H = Gį. Show that the resulting graph H has the desired properties.)
Transcribed Image Text:A graph H is called a subgraph of a graph G if V (H) ≤ V (G) and E(H) ≤ E(G); in this case we write HCG. Prove that every graph G with |E(G)| > 1 edges contains a subgraph H with minimum degree 8(H) > d(G)/2. (Hint: Construct a sequence of subgraphs G = Go 2 G₁ 2. as follows: if G; has a vertex vį, of degree da; (vi) ≤ d(Gi)/2, then obtain G₁+1 from Gį by deleting vį (which includes deleting all edges containing vį); if not, then terminate with H = Gį. Show that the resulting graph H has the desired properties.)
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