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 a vertex u₀ and ending at a vertex
v₀ Let G ⁰ be a graph obtained by adding an edge e₀ joining u₀ and v₀ , so that
G ⁰ − e = G.
(a) Prove that given a semi-Eulerian trail t in G from v₀ to u₀ , it is possible to
construct a Eulerian trail in G₀.
(b) Prove that given an Eulerian trail in G 0 it is possible to construct a semi-Eulerian
trail in G.
(c) Prove Theorem 1.

Expert Solution

a) Let be a semi-Eulerian trail in G from v₀ to u_0. Now, let us add the edge e in G. Then, we can see that this edges makes this trail in a Eulerian trail. Hence, we get a Eulerian trail in G₀

b) Now, let us take an Eulerian trail in G₀. Then we will write the trail as , since it passes each edge once (basically I reset the starting parameter of the trail such that it starts from v_0 and it traverses the edges e at the veyr end). Then, if we take out the edge e from the graph G₀ . Then, we can get a semi Eulerian trail in the graph in G.

c) From part a) and b) we get that a Graph can be converted from Semi Eulerian graph to a Eulerian graph by adding an edge between the endpoints of the semi Eulerian trail. And we can remove an edge from Eulerian graph to make it semi Eulerian. Now, we also know that a Graph is Eulerian if and only if every vertex has even degree. Then, from these information we get that a graph is Semi Eulerian if and only if it has exactly two odd degree vertices. This can be concluded because we are only removing one edge from a Eulerian graph to make it semi Eulerian.

Colby Messinger answered 8 months ago

Question 1 a) Prove that if u and v are distinct vertices of a graph G,...

Question 1 a) Prove that if u and v are distinct vertices of a graph G, there exists a walk from u to v if and only if there exists a path (a walk with distinct vertices) from u to v. b) Prove that a graph is bipartite if and only if it contains no cycles of odd length. Please write legibly with step by step details. Many thanks!

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

An eulerian walk is a sequence of vertices in a graph such that every edge is...

An eulerian walk is a sequence of vertices in a graph such that every edge is traversed exactly once. It differs from an eulerian circuit in that the starting and ending vertex don’t have to be the same. Prove that if a graph is connected and has at most two vertices of odd degree, then it has an eulerian walk.

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.

Problem 8. A bipartite graph G = (V,E) is a graph whose vertices can be partitioned...

Problem 8. A bipartite graph G = (V,E) is a graph whose vertices can be partitioned into two (disjoint) sets V1 and V2, such that every edge joins a vertex in V1 with a vertex in V2. This means no edges are within V1 or V2 (or symbolically: ∀u, v ∈ V1, {u,v} ∉ E and ∀u,v ∈ V2, {u,v} ∉ E). 8(a) Show that the complete graph K2 is a bipartite graph. 8(b) Prove that no complete graph Kn,...

Consider an undirected graph G that has n distinct vertices. Assume n≥3. How many distinct edges...

Consider an undirected graph G that has n distinct vertices. Assume n≥3. How many distinct edges will there be in any circuit for G that contains all the vertices in G? What is the maximum degree that any vertex in G can have? What is the maximum number of distinct edges G can have? What is the maximum number of distinct edges that G can have if G is disconnected?

Let G(V, E,w) be a weighted undirected graph, where V is the set of vertices, E...

Let G(V, E,w) be a weighted undirected graph, where V is the set of vertices, E is the set of edges, and w : E → R + is the weight of the edges (R + is the set of real positive numbers). Suppose T(G) is the set of all minimum spanning trees of G and is non-empty. If we know that the weight function w is a injection, i.e., no two edges in G have the same weight, then:...

. Provide a weighted directed graph G = (V, E, c) that includes three vertices a,...

. Provide a weighted directed graph G = (V, E, c) that includes three vertices a, b, and c, and for which the maximum-cost simple path P from a to b includes vertex c, but the subpath from a to c is not the maximum-cost path from a to c

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

Instructions: Question 1: A directed graph G consists of • vertices (V ) (also called nodes)...

Instructions: Question 1: A directed graph G consists of • vertices (V ) (also called nodes) • edges (E) (also called links). An edge can have extra data. The number of nodes |V | and the number of edges |E| are denoted by n and m respectively. An undirected graph is like a directed graph except the edges do not have direction. Some more definitions: • An undirected graph is connected if there exists a path between all u, v...

Question