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- Prove that a simple 2-connected graph G with at least four vertices is 3-connected if and only if for every triple (x, y, z) of distinct vertices and any edge e not incident with y, G has an x, z-path through e that does not contain y.Prove : for r belongs to Z+, every r connected graph on an even number of vertices with no induced subgraph isomorphic to k1,r+1 has a 1-factor. Show that this is not true if you replace r connected by r edge connectedGive an upper bound on the number e of edges of G in terms of n and g if G is a connected plane graph with n vertices and girth g.
- Let G be a simple connected graph with n vertices and 1/2(n-1)(n-2)+2 edges. Use Ore's theorem to prove that G is Hamiltonian.Hey, The condensation of a graph G with k strong coherence components G1 =.(V1 , E1 ), . . . , Gk = (Vk , Ek )is the reduction of the original graphto its strong coherence components. In this case, the coherence components are combined into one node each in the condensation. The condensation to G is thus the graph G↓=({V1,...,Vk},E),where(Vi,Vj)∈E ⇔i̸=j∧∃u∈Vi,v∈Vj:(u,v)∈E holds. what is the Kondensation G↓ of the graph in the picture? Thank you in advance!Theorem 3.5 states the following: Let G be a loopless graph with at least three vertices, and no isolated vertices. Then G is 2-connected if and only if, for every pair {e, f} of edges of G, there is a cycle of G that contains both e and f.
- Prove that every connected planar graph with less than 12 vertices has a vertex of degree at most 4. [Hint: Assume that every vertex has degree at least 5 to obtain a lower bound on e (together with the upper bound on e in the corollary) that implies v ≥ 12.]Let G be a connected graph of order n = 4 and let k be an integer with 2 ≤ k ≤ n − 2. Prove that if G is not k-connected, then G contains a vertex-cut U with |U| = k − 1 and if it is not k-edge-connected, then G contains an edge-cut X with |X| = k − 116. Let G be a graph with n vertices , t of which have degree K and the others have degree K+1 ,prove that t = (K+1)n-2e , e is number of edges in G .
- Let G be a connected graph with at least one edge and F ⊆ E(G) be an edge cut. Prove that F is a minimal edge cut if and only if G − F contains exactly two connected components.How can I prove "A graph is k-colourable if and only if it is k-partite.”?a) List all the odd vertices of the graph.b) According to Euler’s Theorem, does the graph have an Eulerian circuit? Howdo you know?c) According to Euler’s Theorem, does the graph have an Eulerian path? Howdo you know? What is the difference between a Hamiltonian path and an Eulerian path? A person starting in Columbus must-visit Great Falls, Odessa, andBrownsville (although not necessarily in that order), and then return home toColumbus in one car trip. The road mileage between the cities is shown Columbus Great Falls Odessa Brownsville Columbus --- 102 79 56 Great Falls 102 --- 47 69 Odessa 79 47 --- 72 Brownsville 56 69 72 --- Draw a weighted graph that represents this problem in the space below. Use the first letter of the city when labeling each vertex. Find the weight (distance) of the Hamiltonian circuit formed using the nearest neighbor algorithm. Give the vertices in the circuit in the order they are visited in…