Math Problem Statement

A B C D E F [Graphs generated by this script: setBorder(60,30,60,30);initPicture(-10,10,-10,10); fontsize=14;fontstyle='none';marker=null;line([10,0],[-10,1.2246467991474E-15]);line([10,0],[-5,-8.6602540378444]);line([5,8.6602540378444],[-5,8.6602540378444]);line([5,8.6602540378444],[-10,1.2246467991474E-15]);line([5,8.6602540378444],[-5,-8.6602540378444]);line([-5,8.6602540378444],[-10,1.2246467991474E-15]);line([-5,8.6602540378444],[-5,-8.6602540378444]);line([-5,8.6602540378444],[5,-8.6602540378444]);line([-10,1.2246467991474E-15],[-5,-8.6602540378444]);fontbackground='white';fontfill='blue';text([10,0],'A','belowright');dot([10,0]);fontfill='blue';text([5,8.6602540378444],'B','aboveright');dot([5,8.6602540378444]);fontfill='blue';text([-5,8.6602540378444],'C','aboveleft');dot([-5,8.6602540378444]);fontfill='blue';text([-10,1.2246467991474E-15],'D','aboveleft');dot([-10,1.2246467991474E-15]);fontfill='blue';text([-5,-8.6602540378444],'E','belowleft');dot([-5,-8.6602540378444]);fontfill='blue';text([5,-8.6602540378444],'F','belowright');dot([5,-8.6602540378444]);]

Find an Euler path for the graph. Enter your response as a sequence of vertices in the order they are visited, for example, ABCDEA

Solution

To find an Euler path in a graph, we need a path that visits every edge exactly once. This is possible if and only if the graph has exactly 0 or 2 vertices of odd degree. First, let us analyze the graph:

Step 1: Degree of vertices

  • A: Connected to B, D, and F → degree 3 (odd).
  • B: Connected to A, C, and F → degree 3 (odd).
  • C: Connected to B, D, and E → degree 3 (odd).
  • D: Connected to A, C, and E → degree 3 (odd).
  • E: Connected to C, D, and F → degree 3 (odd).
  • F: Connected to A, B, and E → degree 3 (odd).

All vertices have an odd degree, meaning no Euler path exists because more than 2 vertices have an odd degree.

Conclusion

It is not possible to find an Euler path for this graph.

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Math Problem Analysis

Mathematical Concepts

Graph Theory
Eulerian Paths

Formulas

Euler Path Condition: A graph has an Euler path if and only if it has exactly 0 or 2 vertices of odd degree.

Theorems

Euler's Theorem

Suitable Grade Level

Grades 10-12

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