The Pr¨ufer code 
: T → W defined by equations (1.3) and (1.4) is a bisection. Figure 1.6 shows some trees and their Pr¨ufer codes for n = 6 (one for each isomorphism class, see Exercise 4.1.6). Let G be a connected multi graph having exactly 2k vertices of odd degree (k 
0). Then the edge set of G can be partitioned into k trails. The line graph L(G) of a graph G has as vertices the edges of G; two edges of G are adjacent in L(G) if and only if they have a common vertex in G. For example, the line graph of the complete graph 
is the triangular graph 
.