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

Expert Solution

Colby Messinger answered 1 year ago

Prove a connected simple graph G with 16 vertices and 117 edges is not Eulerian.

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)

Show that a graph without isolated vertices has an Eulerian walk if and only if it...

Show that a graph without isolated vertices has an Eulerian walk if and only if it is connected and all vertices except at most two have even degree.

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)

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...

Given a directed graph, prove that there exists an Eulerian cycle that is also a hamiltonian...

Given a directed graph, prove that there exists an Eulerian cycle that is also a hamiltonian cycle if and only if the graph is a single cycle.

9. Let G be a bipartite graph and r ∈ Z>0. Prove that if G is...

9. Let G be a bipartite graph and r ∈ Z>0. Prove that if G is r-regular, then G has a perfect matching.1 10. Let G be a simple graph. Prove that the connection relation in G is an equivalence relation on V (G)

Let G be a bipartite graph and r ∈ Z>0. Prove that if G is r-regular,...

Let G be a bipartite graph and r ∈ Z>0. Prove that if G is r-regular, then G has a perfect matching. HINT: Use the Marriage Theorem and the Pigeonhole Principle. Recall that G is r-regular means every vertex of G has degree

1. Let G be a k-regular bipartite graph. Use Corollary 3.1.13 to prove that G can...

1. Let G be a k-regular bipartite graph. Use Corollary 3.1.13 to prove that G can be decomposed into r-factors iff r divides k.

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.

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