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I-regular rotation of the infinite complete graph with countable vertex set is one in induced circuit is finite of kngth 1. I-regular rotations are exhibited for all 1 a 3. which Triangular rotations of finite graphs have been extensively studied. In particular, finding such rotations for complete graphs is instrumental in the proof of the Heawood conjecture.. This leads to the question of finding triangular rotations of infinite graphs. Ringe4 ([3], Chap. 12) posed the problem of constructing a triangular rotation fork,, the graph with countabie vertex set in which each pair of distinct vertices is adjacent. The regularity of triangular rotations suggests the following generalization. Let an I-regular rotation of a graph be one for which all the induced circuits have length 1. In this paper, Z-regular rotations of K, are constructed for all 1 3 3.
Let S be a countably infinrte set. A rotario~r 4 of K, is an assignment to each i E S of a cyclic permutation 4i acting on S -{i}. For distinct i and j in S, the circuit from i to j induced by a given fotation 4 is the sequence obtained as follows. Let &P=iaP= j. For k >1, uk = 4at.,(~k -2). For k CO, ak = &i+,(ak+& Let 1 be the smallest positive integer, if one exists, such that al = i and al+ 1 = j. Then the circuit is said to be finite of length I. If no such I exists the circuit is said to be infinite. If for every djstirxt i and j in S the circuit from i to j is of length I, 4 witl be eailed ic -regular.
Suppose I' is a countably L%ite group, fvhich for convenience will be assumed to be abelian+ L& cr be a cyclic permutation of r -{O}, where 0 is the identity in r.
'I%en the assignment to each y E 1" of the permutation 4: delined via @T(S) = y + ~$8 -y ), S# y, is easily checked to constitute a rotation 4b" of Km. 4 a is said to be generated from CT by the addirk rule. cr witJ be said to satisfy At(p) if the circuit from 0 to p induced by 4" is of length 1. Suppose CT satisfies A, (0) and the circuit from 0 to fl is au = O,a, = @, uzta3,. . ,at = 0. Then by definition & = t#i~_*(ak"2) = i&k-$ + +&-a -Qli-1) &hich implies ak +cr = at._1 4~ CM +t ~(a~+ + cx -(tzkwI + a))= 4~k__1+Q{u4-2 + cw) for any (Y E r. Thus if c satisfies;
,& (@)$ then for any CY E r, the circuit induced from CT to a + p is of length J. irr $aq, if CI satisfies AI(p) for at1 0 E f -{O}, then 4" is 1 -regular.