1$ and the statement is true for smaller values of $r-p$. Just like in the previous section, we can assume that $\sigma_k\ne\sigma_p$ for $k=p+1,\dots,r-1$, and $\sigma_p=\{1,N\}$, i. e. $i_p=1$, $j_p=N$. The useful, short form of the Connecting Path Formula for the new advance $A_r$ in terms of the earlier advances is now $$ \aligned A_r\cdot\left(v_1^-(t_r)-v_N^-(t_r)\right)&= A_p\cdot\left(v_1^+(t_p)-v_N^+(t_p)\right) \\ +A_{\{1,2\}}\cdot\sum\Sb k\\ p0$ whereas the remaining $N-1$ ones are point particles, the tree of interactions is star-like, i. e. $\Cal E=\{\{1,i\}: 2 \le i \le N\}$. In this case $D=N-2$, and our result just requires the constraint $\nu \ge N$ for the K-property, thus we can not improve on Sim\'anyi's lower bound. 2. On the other hand, it is easy to see that the arguments used to prove Theorems C and D are also applicable to an arbitrary connnected graph of interaction. Consequently, these arguments combined with the results of [Sim(1992)] provide that, in general, a system of $N$ balls with a connected graph of interactions has open ergodic components whenever $\nu\ge N-1$. \bigskip \bigskip \Refs \widestnumber\key{B-P-L-X(1999)} \ref\key B-L-P-S(1992) \by L. Bunimovich, C. Liverani, A. Pellegrinotti, Yu. Sukhov \paper Special Systems of Hard Balls that Are Ergodic \jour Commun. Math. Phys. \vol 146 \pages 357-396 (1992) \endref \ref\key B-S(1973) \by L. A. Bunimovich, Ya. G. Sinai \paper The fundamental theorem of the theory of scattering billiards \jour Math. USSR-Sb. \vol 19 \pages 407-423 (1973) \endref \ref\key C-H(1995) \by N. I. Chernov, C. Haskell \paper Nonuniformly hyperbolic K-systems are Bernoulli \jour to appear in Ergodic Theory and Dynamical Systems \endref \ref\key G(1981) \by G. Galperin \paper On systems of locally interacting and repelling particles moving in space \jour Trudy MMO \vol 43 \pages 142-196 (1981) \endref \ref\key H-W(1941) \by W. Hurewicz, H. Wallman \paper Dimension Theory \book Princeton, 1941 \endref \ref\key K-S(1986) \by A. Katok, J.-M. Strelcyn \paper Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities \jour Lecture Notes in Mathematics \vol 1222 \pages Springer, 1986 \endref \ref\key K-S-Sz(1989) \by A. Kr\'amli, N. Sim\'anyi, D. Sz\'asz \paper Ergodic Properties of Semi--Dispersing Billiards I. Two Cylindric Scatterers in the 3--D Torus \jour Nonlinearity \vol 2 \pages 311--326 (1989) \endref \ref\key K-S-Sz(1990) \by A. Kr\'amli, N. Sim\'anyi, D. Sz\'asz \paper A ``Transversal'' Fundamental Theorem for Semi-Dis\-pers\-ing Billiards \jour Commun. Math. Phys. \vol 129 \pages 535--560 (1990) \endref \ref\key K-S-Sz(1991) \by A. Kr\'amli, N. Sim\'anyi, D. Sz\'asz \paper The K--Property of Three Billiard Balls \jour Annals of Mathematics \vol 133 \pages 37--72 (1991) \endref \ref\key K-S-Sz(1992) \by A. Kr\'amli, N. Sim\'anyi, D. Sz\'asz \paper The K--Property of Four Billiard Balls \jour Commun. Math. Phys. \vol 144 \pages 107-148 (1992) \endref \ref\key O-W(1995) \by D. Ornstein, B. Weiss \paper On the Bernoulli Nature of Systems with Some Hyperbolic Structure \jour Manuscript, pp. 23 (1995) \endref \ref\key Sim(1992) \by N. Sim\'anyi \paper The K-property of $N$ billiard balls I \jour Invent. Math. \vol 108 \pages 521-548 (1992) \moreref \paper II. \jour ibidem \vol 110 \pages 151-172 (1992) \endref \ref\key Sin(1963) \by Ya. G. Sinai \paper On the Foundation of the Ergodic Hypothesis for a Dynamical System of Statistical Mechanics \jour Soviet Math. Dokl. \vol 4 \pages 1818-1822 (1963) \endref \ref\key Sin(1970) \by Ya. G. Sinai \paper Dynamical Systems with Elastic Reflections \jour Russian Math. Surveys \vol 25:2 \pages 137-189 (1970) \endref \ref\key Sin(1985) \by Ya. G. Sinai \paper Hydrodynamic Limit Transition for a Quasi-One-Dimensional System \jour Tur\'an Memorial Lectures, Budapest \pages manuscript, 1985 \endref \ref\key S-Ch(1985) \by Ya. G. Sinai, N.I. Chernov \paper Ergodic properties of certain systems of 2--D discs and 3--D balls \jour manuscript \endref \ref\key S-Ch(1987) \by Ya. G. Sinai, N.I. Chernov \paper Ergodic properties of certain systems of 2--D discs and 3--D balls \jour Russian Math. Surveys \vol (3) 42 \pages 181-207 (1987) \endref \ref\key Sz(1995) \by D. Sz\'asz \paper Boltzmann's Ergodic Hypothesis, a Conjecture for Centuries? \jour Studia Sci. Math. Hung \pages to appear \endref \ref\key V(1979) \by L. N. Vaserstein \paper On Systems of Particles with Finite Range and/or Repulsive Interactions \jour Commun. Math. Phys. \vol 69 \pages 31-56 (1979) \endref \endRefs \enddocument