GRAPH THEORY PLEASE ANSWER FULLY AND COMPLETELY Draw a conn

GRAPH THEORY - PLEASE ANSWER FULLY AND COMPLETELY

Draw a connected graph on five vertices (if possible) that has the following properties. Your graphs do not have to be simple graphs. For each graph write out the Hamilton cycle or path and Eulerian circuit or trail. (a) A Hamilton cycle and an Eulerian circuit. (b) A Hamilton cycle and Eulerian trail, but no Eulerian circuit. (c) A Hamilton cycle, but neither an Eulerian circuit nor Eulerian trail. (d) An Eulerian circuit and a Hamilton path, but no Hamilton cycle. (e) An Eulerian circuit, but neither a Hamilton cycle nor a Hamilton path.

Solution

If your points area unit A, B, C, D, E then the sides AB, BC, CD, DE, Ea clearly give one in every of the tiniest graphs that has each a Eulerian cycle and Hamiltonian cycle.

Add another edge, as an example AC, associate degreed you\'ve got a graph that has identical Hamiltonian cycle however not an Eulerian cycle, since A associate degreed C have an odd variety of edges.

Alternatively, the sides AB, BC, CA, AD, DE, EA, give a Eulerian cycle however not a Hamiltonian cycle, since you\'ll solely get from B to D via A, and you can not come back to while not visiting A once more