Advanced Concepts in Euler Trails and Circuits Quiz

1) An Euler trail in an undirected graph is:

A closed walk that visits every vertex exactly once

A walk that uses every edge at least once

A walk that uses every edge exactly once (not necessarily closed)

Any maximal path

An Euler trail uses each edge exactly once; start/end may differ.

Explanation

An Euler trail uses each edge exactly once; start/end may differ.

2) Every graph that has an Euler circuit automatically has at least one Euler trail.

True

False

An Euler circuit is a special closed Euler trail.

Explanation

An Euler circuit is a special closed Euler trail.

3) A connected graph has vertex degrees 1,1,2,2,2. Which statement is true?

It has no Euler trail

It has an Euler circuit

It has an Euler trail but not an Euler circuit

It must be a tree

Exactly two vertices have odd degree → Euler trail but no circuit.

Explanation

Exactly two vertices have odd degree → Euler trail but no circuit.

4) A connected graph has degrees 4,4,4,4. Which statement is correct?

It has an Euler circuit

It has an Euler trail but not a circuit

It has no Euler trail

It cannot exist

All degrees even → Eulerian.

Explanation

All degrees even → Eulerian.

5) For an undirected graph, having all vertices of even degree is sufficient but not necessary for an Euler trail.

True

False

A trail exists with exactly two odd-degree vertices too.

Explanation

A trail exists with exactly two odd-degree vertices too.

6) Which condition characterizes a directed Euler circuit?

Graph is connected and each vertex has out-degree 1

Underlying graph is connected and indegree=outdegree for all vertices

All vertices have even total degree

Graph is acyclic and strongly connected

Directed Euler circuits require indegree=outdegree and connectivity.

Explanation

Directed Euler circuits require indegree=outdegree and connectivity.

7) In a directed graph with 10 edges, the sum of all out-degrees is:

5

10

20

Depends

Each edge contributes 1 to out-degree.

Explanation

Each edge contributes 1 to out-degree.

8) A connected undirected graph has an Euler trail but not a circuit. How many vertices have odd degree?

0

1

2

3

Euler trail but no circuit ⇔ exactly two odd-degree vertices.

Explanation

Euler trail but no circuit ⇔ exactly two odd-degree vertices.

9) In any finite undirected graph, the number of vertices with odd degree is always even.

True

False

Handshake lemma forces even number of odd-degree vertices.

Explanation

Handshake lemma forces even number of odd-degree vertices.

10) Does the path graph P5 admit an Euler circuit?

Yes, exactly one

Yes, many

No, but has Euler trail

No trail

P5 has two odd endpoints → Euler trail but no circuit.

Explanation

P5 has two odd endpoints → Euler trail but no circuit.

11) Given the multigraph description, does it have an Euler trail?

Yes, and also a circuit

Yes, but not an Euler circuit

No Euler trail

Only if a new edge is added

Exactly two vertices have odd degree → Euler trail but not circuit.

Explanation

Exactly two vertices have odd degree → Euler trail but not circuit.

12) Which real-world problem fits Eulerian modification?

Traveling Salesman

Chinese Postman

Minimum Spanning Tree

Network Flow

Chinese Postman seeks route using all edges with minimal repeats.

Explanation

Chinese Postman seeks route using all edges with minimal repeats.

13) Euler circuit test condition?

True

False

All vertices even degree and connected.

Explanation

All vertices even degree and connected.

14) A connected undirected graph has degrees 2,2,2,2,2,2. Which is true?

No Euler trail

Exactly one Euler circuit

At least one Euler circuit

Euler trail but no circuit

All even degrees → Euler circuit exists.

Explanation

All even degrees → Euler circuit exists.

15) A connected graph has 12 edges and exactly two odd vertices. Sum of degrees?

12

22

24

Depends

Sum of degrees = 2|E| = 24.

Explanation

Sum of degrees = 2|E| = 24.