In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. Dirac's Theorem - If G is a simple graph with n vertices, where n ≥ 3 If deg(v) ≥ {n}/{2} for each vertex v, then the graph G is Hamiltonian graph. . C4 (=K2,2) is a cycle of four vertices, 0 connected to 1 connected to 2 connected to 3 connected to 0. If there exists a walk in the connected graph that visits every vertex of the graph exactly once (except starting vertex) without repeating the edges and returns to the starting vertex, then such a walk is called as a Hamiltonian circuit. If H is either an edge or K4 then we conclude that G is planar. Vertex set: Edge set: The graph is clearly Eularian and Hamiltonian, (In fact, any C_n is Eularian and Hamiltonian.) A complete graph with 8 vertices would have = 5040 possible Hamiltonian circuits. Half of the circuits are duplicates of other circuits but in reverse order, leaving 2520 unique routes. Definition. Based on these results we define socalled K4-closures of G. We give infinite classes of graphs with small maximum degree and large diameter, and with many vertices of degree two having complete K4-closures. If you label 0 and 2 as "A", and 1 and 3 as "B", you can see that the graph connects only A's to B's, and not A's to A's or B's to B's. A complete graph K4. Toughness and harniltonian graphs It is easy to see that every cycle is 1-tough. The hamiltonian path graph H(F) of a graph F is that graph having the same vertex set as F and in which two vertices u and v are adjacent if and only if F contains a hamiltonian u − v path. 2. 1. Explicit descriptions Descriptions of vertex set and edge set. Every complete graph has a Hamilton circuit. Every hamiltonian graph is 1-tough. First, in response to a conjecture of Chartrand, Kapoor and Nordhaus, a characterization of nonhamiltonian graphs isomorphic to their hamiltonian path graphs is presented. Based on these results we define socalled K4-closures of G. We give infinite classes of graphs with small maximum degree and large diameter, and with many vertices of degree two having complete K4-closures. Hamiltonian Path Examples- Examples of Hamiltonian path are as follows- Hamiltonian Circuit- Hamiltonian circuit is also known as Hamiltonian Cycle.. Hamiltonian walk in graph G is a walk that passes through each vertex exactly once. While this is a lot, it doesn’t seem unreasonably huge. If clock-wise and anti-clockwise cycle is same then we divide total permutations with 2. for example two cycles 123 and 321 both are same because they are reverse of each other. If e is not less than or equal to 3n – 6 then conclude that G is nonplanar. 1. Hamiltonian graph - A connected graph G is called Hamiltonian graph if there is a cycle which includes every vertex of G and the cycle is called Hamiltonian cycle. It is also sometimes termed the tetrahedron graph or tetrahedral graph.. 1. 1 is 1-connected but its cube G3 = K4 -t- K3 is not Z -tough. The complete graph with 4 vertices is written K4, etc. The graph G in Fig. As a consequence, a claw-free graph G is hamiltonian if and only if G+uv is hamiltonian, where u, u is a K4-pair. The first three circuits are the same, except for what vertex Actualiy, (G 3) = 3; using Proposition 1.4, we conclude that t(G3y< 3. n t Fig. KW - IR-29721. This observation and Proposition 1.1 imply Proposition 2.1. This graph, denoted is defined as the complete graph on a set of size four. H is non separable simple graph with n 5, e 7. 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