Let \((W,S,\Gamma)\) be a Coxeter system, where \(W\) is a Coxeter group, \(S=\{s_1,s_2,\dots,s_r\}\) a distinguished generating set, and \(\Gamma\) the corresponding Coxeter graph. By a Coxeter element \(w\in W\), one means a product \(s_{i_1}s_{i_2}\cdots s_{i_r}\) with \(i_1,i_2,\dots,i_r\) a permutation of \(1,2,\dots,r\). Let \(C(W)\) be the set of all the Coxeter elements in \(W\). The conjugacy relation in \(C(W)\) is relatively less known except for the case where \(\Gamma\) is a tree; in the latter case, \(C(W)\) is entirely contained in a single \(W\)-conjugacy class [see \textit{J.~E.~Humphreys}, ``Reflection groups and Coxeter groups'', Cambridge Studies in Advanced Mathematics (1992; Zbl 0768.20016)]. In this paper the author mainly considers the case where \(\Gamma\) contains just one circle. The author first introduces the concept of ss-equivalence in \(C(W)\). Each ss-class of \(C(W)\) is contained in some \(W\)-conjugacy class. The author shows in Proposition 2.3 that the study of ss-classes in \(C(W)\) with \(\Gamma\) containing just one circle can be reduced to the case where \(\Gamma\) is itself a circle. The author describes all the ss-classes of \(C(W)\) when \(\Gamma\) is a circle. Furthermore, the author shows that the ss-equivalence relation in \(C(W)\) is actually the same as the \(W\)-conjugacy relation in the following two special cases: when \(\Gamma\) is a three-multiple circle and when \(\Gamma\) is a circle with three nodes. An explicit formula is given for the characteristic polynomial of a Coxeter element in the natural reflection representation of \(W\) when \(\Gamma\) is a circle. The author also gives answers to some questions raised by \textit{A.~J.~Colemen} [Invent. Math. 95, No. 3, 447-477 (1989; Zbl 0679.17008)] and extends some results of \textit{M.~Geck} and \textit{G.~Pfeiffer} [Adv. Math. 102, No. 1, 79-94 (1993; Zbl 0816.20034)] concerning the conjugacy relation in Coxeter groups.