Hamilton Cycle on the Wheel Graph

Abstract

This article discusses the existence of the Hamilton cycle in the wheel graph by constructing steps to find the existence of the Hamilton cycle. A graph that has a Hamilton cycle is called a Hamilton graph, A circle graph is a graph where each vertex has a degree of two, denoted by C_n. A graph obtained by adding a central vertex to a circle graph and connecting it to all the vertices of the circle graph is called a wheel graph, denoted by W_n . If the wheel graph W_n  has m where m is the number of  that replaces each point in W_n then it can be denoted by W^m_n . Then, in the wheel graph W^m_n  is the number of outermost points of W^m_n  added to 1 point located in the center. Based on the construction, it is found that there is a Hamilton cycle in the wheel graph. In the wheel graph W_n contains Hamilton cycle for n>=3. Furthermore, the wheel graph W^m_n  also contains Hamilton cycle for n>=3 and m>=1, but the image of the wheel graph W^m_n is only perfectly drawn for n=2k where k is an integer. This is because there are colliding edges in the wheel graph for n=2k-1 where k is an integer.