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.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Let G be a simple graph. G is said to be maximal planar if it is...

Let G be a simple graph. G is said to be maximal planar if it is planar and the addition of any new edge to G results in a (simple) non-planar graph. Examples of maximal non-planar graphs are K4 , K5 minus an edge, and K3,3 minus an edge. (a) Show that a maximal planar graph is connected. (b) Show that a maximal planar graph of order ≥3 has no bridges. (c) Show that every face of a maximal planar...

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

Let G be a simple graph. Prove that the connection relation in G is an equivalence...

Let G be a simple graph. Prove that the connection relation in G is an equivalence relation on V (G)

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?)

Let G be a graph. prove G has a Eulerian trail if and only if G...

Let G be a graph. prove G has a Eulerian trail if and only if G has at most one non-trivial component and at most 2 vertices have odd degree

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 A be incidence matrix of graph G. prove if G has a cycle, then null...

Let A be incidence matrix of graph G. prove if G has a cycle, then null space of transpose of A is not {0} (there exists non-trivial solution of (A^T)y=0)

prove by induction that for any simple connected graph G, if G has exactly one cycle...

prove by induction that for any simple connected graph G, if G has exactly one cycle then G has the same number of edges and nodes

1) a) Let k ≥ 2 and let G be a k-regular bipartite graph. Prove that G...

1) a) Let k ≥ 2 and let G be a k-regular bipartite graph. Prove that G has no cut-edge. (Hint: Use the bipartite version of handshaking.) b) Construct a simple, connected, nonbipartite 3-regular graph with a cut-edge. (This shows that the condition “bipartite” really is necessary in (a).) 2) Let F_n be a fan graph and Let a_n = τ(F_n) where τ(F_n) is the number of spanning trees in F_n. Use deletion/contraction to prove that a_n = 3a_n-1 - a_n-2...

Let G a graph of order 8 with V (G) = {v1, v2, . . ....

Let G a graph of order 8 with V (G) = {v1, v2, . . . , v8} such that deg vi = i for 1 ≤ i ≤ 7. What is deg v8? Justify your answer. Please show all steps thank you

Subjects