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 2 years 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)

GRAPH THEORY Prove/Show that a connected Graph G is not separable if and only if it...

GRAPH THEORY Prove/Show that a connected Graph G is not separable if and only if it is nonseparable. Definitions for Reference: A connected Graph G is called nonseparable if it has no cut vertices (A vertex v in a connected graph G is caled a cut vertex if G-v is disconnected) A connected graph G is called separable if there exist subgraphs H1, H2 ⊂ G. with E(H1) ∪ E(H2) = E(G) and E(H1) ∩ E(H2) = ∅. V (H1)...

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...

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)

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.

A graph is G is semi-Eulerian if there are distinct vertices u, v ∈ V (G),...

A graph is G is semi-Eulerian if there are distinct vertices u, v ∈ V (G), u =v such that there is a trail from u to v which goes over every edge of G. The following sequence of questions is towards a proof of the following: Theorem 1. A connected graph G is semi-Eulerian (but not Eulerian) if and only if it has exactly two vertices of odd degree. Let G be semi-Eulerian with a trail t starting at...

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

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