CIS. –trans conformationThe distribution of the O—Si—Si—O torsion angles during the coolingcycle of HP is shown in Fig. 8. At 1500 K, all the torsion angles in plane are 60◦, which signifies the trans- conformation. In contrast, all the angles out of plane are 0◦, i.e. in the cis-conformation. When the temperaturedecreases to 900 K, the in-plane anglessplit into two peaks, although no change in the out-of-plane angles is observed. At 10 K, the in- plane angles are split into three peaks. The in-plane angles suggest that the trans-conformation deviates from the idealvalue of 60◦. In contrast, the out-of-plane angles suggest that there is only a smalldeviation from the ideal cis value of 0◦. Next, the distribution of O—Si—Si—O torsion angles during the heating of MX1 is shown in Fig. 9. The starting distribution at 10 K has more peaks and the shape is more distorted than that of HP. The out-of-plane peak deviates from the ideal value of 0◦. As the temperature increases up to 300 K, the number of peaks for the in-plane angles reduces to two. At 700 K, the out-of-plane peak moves to the ideal value of 0◦. Finally, above 1100 K, the distribution reverts to thesame as that of HP.

CIS d-graphs and ∆-conjectureDefinition 6 A d-graph 5 = (V ; E1, . . . , Ed) is a complete graph whose edges are arbitrarily partitioned into d subsets (colored with d colors). Graph Gi = (V, Ei) is called the ith chromatic component of 5, where i ∈ [d] = {1, . . . , d}.In case d = 2 a d-graph is just a graph, or more precisely, a pair: a graph and its complement. Thus, d-graphs can be viewed as a generalization of graphs.by £ = {Si | i ∈ [d]} the obtained set-family; furthermore, let S =di=1Si.Choose a maximal independent set Si ⊆ V in every graph Gi andTdenoteObviously, |S| ≤ 1 for every £; indeed, if v, v′ ∈ S then (v, v′) /∈ Ei for all