Let n ≥ 3. Show that if G is a graph with the same chromatic polynomial...

Let n ≥ 3. Show that if G is a graph with the same chromatic polynomial as Cn, then G is isomorphic to Cn. (Hint: Use induction. What kind of graph must G − e be for any edge e? Why?)

Expert Solution

Colby Messinger answered 2 years ago

Let G be a simple planar graph with no triangles. (a) Show that G has a...

Let G be a simple planar graph with no triangles. (a) Show that G has a vertex of degree at most 3. (The proof was sketched in the lectures, but you must write all the details, and you may not just quote the result.) (b) Use this to prove, by induction on the number of vertices, that G is 4-colourable.

Let G = R\{0} and N = (0,∞). Show that G/N is isomorphic to the multiplicative...

Let G = R\{0} and N = (0,∞). Show that G/N is isomorphic to the multiplicative group {1, −1}.

Let Q:= {1,2,...,q}. Let G be a graph with the elements of Q^n as vertices and...

Let Q:= {1,2,...,q}. Let G be a graph with the elements of Q^n as vertices and an edge between (a1,a2,...,an) and (b1,b2,...bn) if and only if ai is not equal to bi for exactly one value of i. Show that G is Hamiltonian.

Let G be a simple undirected graph with n vertices where n is an even number....

Let G be a simple undirected graph with n vertices where n is an even number. Prove that G contains a triangle if it has at least (n^2 / 4) + 1 edges using mathematical induction.

let G be a simple graph. show that the relation R on the set of vertices...

let G be a simple graph. show that the relation R on the set of vertices of G such that URV if and only if there is an edge associated with (u,v) is a symmetric irreflexive relation on G

A maximal plane graph is a plane graph G = (V, E) with n ≥ 3...

A maximal plane graph is a plane graph G = (V, E) with n ≥ 3 vertices such that if we join any two non-adjacent vertices in G, we obtain a non-plane graph. a) Draw a maximal plane graphs on six vertices. b) Show that a maximal plane graph on n points has 3n − 6 edges and 2n − 4 faces. c) A triangulation of an n-gon is a plane graph whose infinite face boundary is a convex n-gon...

Show that any graph with n vertices and δ(G) ≥ n/2 + 1 has a triangle.

Let G be a connected graph and let e be a cut edge in G. Let...

Let G be a connected graph and let e be a cut edge in G. Let K be the subgraph of G defined by: V(K) = V(G) and E(K) = E(G) - {e} Prove that K has exactly two connected components. First prove that e cannot be a loop. Thus the endpoint set of e is of the form {v,w}, where v ≠ w. If ṽ∈V(K), prove that either there is a path in K from v to ṽ, or...

Let f be an irreducible polynomial of degree n over K, and let Σ be the...

Let f be an irreducible polynomial of degree n over K, and let Σ be the splitting field for f over K. Show that [Σ : K] divides n!.

Let N be a normal subgroup of the group G. (a) Show that every inner automorphism...

Let N be a normal subgroup of the group G. (a) Show that every inner automorphism of G defines an automorphism of N. (b) Give an example of a group G with a normal subgroup N and an automorphism of N that is not defined by an inner automorphism of G

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