Correction to “An application of graph theory to algebra”
Richard G. Swan · Proceedings of the American Mathematical Society · 1969
PROOF. Let r' be the result of deleting e, e', and X. The theorem holds for r' by induction. Any unicursal path on r' has the form 7172 ... 7r* where each 7ri is a path starting and ending at P but not meeting P between. Clearly n is the number of edges leaving P in r, and so is the same for all paths. Let X be the path ee' from P to P in r. We get all possible unicursal paths on r by starting with such paths on rI and inserting X, getting X7w1 * * * 7r., T.1X .. *nX . . . I 7ri * * * 7nAX. Assuming that e, e' are the last two edges in the chosen ordering of the edges, we have-(71 . . . 7-jiX-i+1 * * * 7rn) =E(7ri . . . 7) Thus E(7r) = (n+1) ,E(r') =0, the first sum being over all unicursal paths on r and the second over such paths on r'. We now consider Case 2 of [1]. If P =A, we can repeat the argument of Case 2 using B and C in place of B and A with only minor modifications. This will be possible provided C#A, but if C=A, the lemma applies with X = B. Suppose now that P = B. Let U be the set of unicursal paths on r starting at A, U' the set of such paths which begin with e, and Ui the set of unicursal paths on ri starting at A. Then the argument of [1, Case 2] shows that U= U'YUUi, a disjoint union. Since the theorem holds for U by what we have just proved, and also for Us, we see that E(r') = 0 where 7r' runs over all elements of U'. But there is a one-to-one correspondence between U' and the set of unicursal paths starting from B, given by ee'e, ... e. elel *.*.* ene. Since n =E-2 is even, e(ee'e1 . . . en) = -e(e' * * * e,e). Therefore the theorem holds in this case also. Finally, we consider Case 3. If there is an edge not meeting P, choose it for e4. Then P A, B and we are done. Suppose every edge meets P. Let P, A1, * * * I A. be the vertices. Then E=2V=2n+2.