Graph That Has A Euler Circuit But Not A Hamiltonian Circuit

Eulerian Circuit Visits each edge exactly once. Starts and ends on same vertex. Is it possible a graph has a hamiltonian circuit but not an eulerian circuit? Here is my attempt based on proof by contradiction Suppose there is a graph G that has a hamiltonian circuit. That means every vertex has at least one neighboring edge. lt-- stuck

i need to give an example of a connected graph with at least 5 vertices that has as an Eulerian circuit, but no Hamiltonian cycle? graph-theory hamiltonian-path Share. Cite. Follow the graph has an Eulerian circuit. Share. Cite. Follow answered Feb 3, 2014 at 2354. Rebecca J. Stones Rebecca J. Stones. 27.2k 2 2 gold badges

Example 12. Give an example of a graph which has a Hamiltonian circuit but not an Euler circuit. A.U MJ 2013 Solution The graph having a Hamiltonian circuit but not an Euler circuit is shown in figure. The Hamiltonian circuit is V 1, V 2, V 4, V 3, V 1. There is no Euler circuit. Example 13.

This graph has an Euler path but not a circuit because it has exactly two odd-degree vertices. One possible path P-S-R-Q-P-R. 3. Determine if the following graph has a Hamiltonian cycle A complete graph K 5 with 5 vertices. A complete graph K n always has a Hamiltonian cycle for n 3. So yes, K 5 has a Hamiltonian cycle.

The full bipartite graph is non-Hamiltonian but has an Eulerian circuit. Step-by-step explanation An Euler circuit starts and ends at the same vertex and uses each vertex exactly once Eulerian circuit. It's worth noting that an Eulerian circuit can visit vertices multiple times. Each vertex is visited exactly once by a Hamiltonian cycle.

One Hamiltonian circuit is shown on the graph below. There are several other Hamiltonian circuits possible on this graph. Notice that the circuit only has to visit every vertex once it does not need to use every edge. This circuit could be notated by the sequence of vertices visited, starting and ending at the same vertex ABFGCDHMLKJEA.

A connected graph G is Eulerian if there is a closed trail which includes every edge of G, such a trail is called an Eulerian trail. Hamiltonian Cycle. A connected graph G is Hamiltonian if there is a cycle which includes every vertex of G such a cycle is called a Hamiltonian cycle. Consider the following examples This graph is BOTH Eulerian

Corollary A graph has an Eulerian circuit if and only if it is connected and all of its vertices have even degree. Note that these conditions are sufficient but not necessary there are graphs that have Hamilton circuits but do not meet these conditions. 92C_692 for example cycle with 6 vertices each vertex has degree 2 and 922lt62

Notice that if a graph has an Eulerian path that is not a circuit it is generally not considered an Eulerian graph, although some authors will call it such. So in any reference you read, be sure to check that definition that is used! The circuit itself, called the Gray Code, is not the only Hamiltonian circuit of the 92n92-cube, but it is