Question: c) Consider the graph below: A B D C (i) Write the degree of each vertex of the graph. (2 marks) (ii) Identify whether the graph contains a Euler circuit. If it does, write down a Euler circuit by listing the vertices in the order they are visited, starting and ending at vertex A. If it does not, explain why. (2 marks) (iii) Identify whether the graph contains a Euler path. If it does, write down a Euler path by listing the vertices in the order they are visited, starting at vertex A. If it does not, explain why. (2 marks) Graph shape: a square with vertices A top-left, B top-right, C bottom-right, and D bottom-left, with edges AB, BC, CD, DA, plus diagonals AC and DB; position_hint=center.

1. **State the problem:** We have a graph with vertices $A$, $B$, $C$, and $D$ arranged as a square with edges $AB$, $BC$, $CD$, $DA$, and diagonals $AC$ and $DB$. We need to find the degree of each vertex, determine if there is a Euler circuit, and determine if there is a Euler path. 2. **Degree of each vertex:** The degree of a vertex is the number of edges incident to it. - Vertex $A$ is connected to $B$, $D$, $C$ (edges $AB$, $DA$, $AC$), so degree of $A$ is $3$. - Vertex $B$ is connected to $A$, $C$, $D$ (edges $AB$, $BC$, $DB$), so degree of $B$ is $3$. - Vertex $C$ is connected to $B$, $D$, $A$ (edges $BC$, $CD$, $AC$), so degree of $C$ is $3$. - Vertex $D$ is connected to $A$, $C$, $B$ (edges $DA$, $CD$, $DB$), so degree of $D$ is $3$. 3. **Euler circuit conditions:** A graph has a Euler circuit if it is connected and every vertex has an even degree. - Here, all vertices have degree $3$, which is odd. - Therefore, the graph does **not** have a Euler circuit. 4. **Euler path conditions:** A graph has a Euler path if it is connected and exactly zero or two vertices have an odd degree. - Here, all four vertices have odd degree ($3$). - Since more than two vertices have odd degree, the graph does **not** have a Euler path. **Final answers:** - (i) Degree of $A=3$, $B=3$, $C=3$, $D=3$. - (ii) No Euler circuit because all vertices have odd degree. - (iii) No Euler path because more than two vertices have odd degree.