Discrete Mathematics..

530 Graph Theory(i)(ii)The compound whose chemical formula is C 3 H 6 has two structuralisomers; draw their graphs. (One of these is propene and the otheris cyclo-propane.)Draw the graphs of the different structural isomers of C 4 H 8 .(Someof these are butenes; others are methyl-cyclo-propane and cyclobutane.)15. The hydrocarbon C 5 H 8 has many structural isomers. We found 27 (wethink!). How many can you find?(They delight in such names as pentyne, methyl-butadiene, methyl-cyclobutene,etc. Several of the isomorphism types do not exist as chemicalcompounds because the stresses in the molecular structure are too great.Of course, the mathematics cannot tell us that—it is the domain of thechemists.)10.5 Planar GraphsExamining the diagrams of graphs above it is apparent that some graphs can berepresented by diagrams drawn in the plane (without any edges crossing) andsome cannot. The object of this section is to examine which graphs can berepresented by plane diagrams. Clearly this question is of potential importancein some of the applications of graph theory, notably in the design of electroniccircuits which can be printed on boards.The question of whether a graph can be represented in the plane is illustrated bythe so-called three utilities problem. Three houses are each to be supplied withthree utilities—electricity, gas and water. The problem is whether the three housescan be supplied without any of the utility lines having to cross. The graph whichmodels this situation is the complete bipartite graph K 3,3 . The three vertices ofone set represent the three utility outlets and the three of the second set representthe three houses. The edges of the graph represent the utility lines; each utility isconnected to each house. In graph-theoretic terms the problem is whether K 3,3can be represented in the plane. Our diagram of K 3,3 (figure 10.5(b)) is not drawnin the plane, but if we were more ingenious perhaps we could have drawn thediagram without the edges crossing.Intuitively, we say a graph is ‘planar’ if it can be represented by a diagram in theplane with no edges crossing. The formal definition is the following.