1. Half of the circuits are duplicates of other circuits but in reverse order, leaving 2520 unique routes. A complete graph with 8 vertices would have = 5040 possible Hamiltonian circuits. 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 walk in graph G is a walk that passes through each vertex exactly once. C4 (=K2,2) is a cycle of four vertices, 0 connected to 1 connected to 2 connected to 3 connected to 0. 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. A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian path that is a cycle.Determining whether such paths and cycles exist in graphs is the Hamiltonian path problem, which is NP-complete. . A complete graph K4. While this is a lot, it doesn’t seem unreasonably huge. The first three circuits are the same, except for what vertex Vertex set: Edge set: 2. 1 is 1-connected but its cube G3 = K4 -t- K3 is not Z -tough. 3. Circular Permutations: The number of ways to arrange n distinct objects along a fixed circle is (n-1)! This graph, denoted is defined as the complete graph on a set of size four. 1. If e is not less than or equal to 3n – 6 then conclude that G is nonplanar. 1. Else if H is a graph as in case 3 we verify of e 3n – 6. Every hamiltonian graph is 1-tough. H is non separable simple graph with n 5, e 7. Toughness and harniltonian graphs It is easy to see that every cycle is 1-tough. K3 has 6 of them: ABCA, BCAB, CABC and their mirror images ACBA, BACB, CBAC. 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. First, in response to a conjecture of Chartrand, Kapoor and Nordhaus, a characterization of nonhamiltonian graphs isomorphic to their hamiltonian path graphs is presented. 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. 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. 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 H is either an edge or K4 then we conclude that G is planar. KW - IR-29721. 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. It is also sometimes termed the tetrahedron graph or tetrahedral graph.. Actualiy, (G 3) = 3; using Proposition 1.4, we conclude that t(G3y< 3. n t Fig. 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. The graph is clearly Eularian and Hamiltonian, (In fact, any C_n is Eularian and Hamiltonian.) 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. As a consequence, a claw-free graph G is hamiltonian if and only if G+uv is hamiltonian, where u,v is a K4-pair. Every complete graph has a Hamilton circuit. Definition. 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