Let G be a connected graph with m 2 vertices of odd degree. Prove that once is m/2. Let G be a connected graph with m 2 vertices of odd degree. Prove that once is m/2.
49.12. Let G be a graph with n 2 2 vertices. a. Prove that if G has at least ("21) +1 edges, then G is connected. b. Show that the result in (a) is best possible; that is, for each n 2 2, prove there is a graph with ("2) edges that is not connected. 49.12. Let G be a graph with n 2 2 vertices. a. Prove that if G has at least ("21) +1 edges, then G is...
7.5 (i) Prove that, if G is a bipartite graph with an odd number of vertices, then G is non-Hamiltonian. (ii) Deduce that the graph in Fig. 7.7 is non-Hamiltonian. Fig. 7.7 (iii) Show that, if n is odd, it is not possible for a knight to visit all the squares of an n chessboard exactly once by knight's moves and return to its starting point.
(a) Let L be a minimum edge-cut in a connected graph G with at least two vertices. Prove that G − L has exactly two components. (b) Let G an eulerian graph. Prove that λ(G) is even.
Let G be a graph in which there is a cycle C odd length that has vertices on all of the other odd cycles. Prove that the chromatic number of G is less than or equal to 5.
(a) Let G be a graph with order n and size m. Prove that if (n-1) (n-2) m 2 +2 2 then G is Hamiltonian. (b) Let G be a plane graph with n vertices, m edges and f faces. Using Euler's formula, prove that nmf k(G)+ 1 where k(G) is the mumber of connected components of G. (a) Let G be a graph with order n and size m. Prove that if (n-1) (n-2) m 2 +2 2 then...
Prove that if G is a connected graph of order n ≥ 2, then the vertices of G can be listed as v1, v2, . . . , vn such that each vertex vi (2 ≤ i ≤ n) is adjacent to some vertex in the set {v1, v2, . . . , vi−1}.
Exercise 5. Let G be a graph in which every vertex has degree at least m. Prove that there is a simple path (i.e. no repeated vertices) in G of length m.
A connected simple graph G has 16 vertices and 117 edges. Prove G is Hamiltonian and prove G is not Eulerian
er (a) Let G be a connected graph and C a non-trivial circuit in G. Prove directly that if an edge ={a, b} is removed from then the subgraph S CG that remains is still connected. Directly' means using only the definitions of the concepts involved, in this case 'connected' and 'circuit'. Hint: If r and y are vertices of G connected by path that includes e, is there an alternative path connecting x to y that avoids e? (b)...
Prove that, if G is a nontrivial connected graph in which every vertex has even degree, then the edge connectivity of G is no less than 2. Prove that, if G is a nontrivial connected graph in which every vertex has even degree, then the edge connectivity of G is no less than 2.