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. First, in response to a conjecture of Chartrand, Kapoor and Nordhaus, a characterization of nonhamiltonian graphs isomorphic to their hamiltonian path graphs is presented. If e is not less than or equal to 3n – 6 then conclude that G is nonplanar. K3 has 6 of them: ABCA, BCAB, CABC and their mirror images ACBA, BACB, CBAC. A complete graph K4. This observation and Proposition 1.1 imply Proposition 2.1. 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. Actualiy, (G 3) = 3; using Proposition 1.4, we conclude that t(G3y< 3. n t Fig. 1. The first three circuits are the same, except for what vertex It is also sometimes termed the tetrahedron graph or tetrahedral graph.. . Definition. This graph, denoted is defined as the complete graph on a set of size four. A complete graph with 8 vertices would have = 5040 possible Hamiltonian circuits. 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. Circular Permutations: The number of ways to arrange n distinct objects along a fixed circle is (n-1)! If H is either an edge or K4 then we conclude that G is planar. C4 (=K2,2) is a cycle of four vertices, 0 connected to 1 connected to 2 connected to 3 connected to 0. KW - IR-29721. 1. 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. Explicit descriptions Descriptions of vertex set and edge set. 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. 3. Hamiltonian Path Examples- Examples of Hamiltonian path are as follows- Hamiltonian Circuit- Hamiltonian circuit is also known as 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. Else if H is a graph as in case 3 we verify of e 3n – 6. The graph is clearly Eularian and Hamiltonian, (In fact, any C_n is Eularian and Hamiltonian.) 2. Every hamiltonian graph is 1-tough. 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. 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. While this is a lot, it doesn’t seem unreasonably huge. Half of the circuits are duplicates of other circuits but in reverse order, leaving 2520 unique routes. Hamiltonian walk in graph G is a walk that passes through each vertex exactly once. Toughness and harniltonian graphs It is easy to see that every cycle is 1-tough. 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. Every complete graph has a Hamilton circuit. 1 is 1-connected but its cube G3 = K4 -t- K3 is not Z -tough. H is non separable simple graph with n 5, e 7. 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. 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