MiRNA-mRNA interaction depends on multiple factors such as 3UTR isoforms, the cell and tissue-specific expression levels of RNA-binding proteins, the sequence context around the mRNA target site, and other mechanisms. Genetic polymorphisms within miRNAs and their target sites appear to be among the most important ones because they influence the mode and outcome of miRNA-mRNA interaction universally and irreversibly. SNP disruption of miRNAs and their binding sites, as well as conformational changes preventing the access of the miRNA to its target site, are adopted as the most credible mechanistic explanations of SNP-mediated effects. The occurrence of multiple SNPs within the same miRNA-binding site implies their combinatorial mode of action. The presence of the repetitive (homologous) binding sites for the same miRNA on its mRNA target may both enhance the miRNA targeting and provide for the backup target site instead of the one disrupted by SNP, thus rescuing the miRNA functionality. While being underexplored, the multiple genetic polymorphisms within the miRNA-binding sites, as well as homologous miRNA-binding sites, may be considered as additional factors influencing miRNA-mediated regulation of gene expression.

In this paper, we investigate partitions of highly symmetrical discrete structures called cycloids. In general, a mixed hypergraph has two types of hyperedges. The vertices are colored in such a way that each C-edge has two vertices of the same color, and each D-edge has two vertices of distinct colors. In our case, a mixed cycloid is a mixed hypergraph whose vertices can be arranged in a cyclic order, and every consecutive p vertices form a C-edge, and every consecutive q vertices form a D-edge in the ordering. We completely determine the maximum number of colors that can be used for any $p\ge 3$ and any $q\ge 2$. We also develop an algorithm that generates a coloring with any number of colors between the minimum and maximum. Finally, we discuss the colorings of mixed cycloids when the maximum number of colors coincides with its upper bound, which is the largest cardinality of a set of vertices containing no C-edge.

In colorings of some block designs, the vertices of blocks can be thought of as hyperedges of a hypergraph $\mathcal{H}$ that can be placed on a circle and colored according to some rules that are related to colorings of circular mixed hypergraphs. A mixed hypergraph $\mathcal{H}$ is called circular if there exists a host cycle on the vertex set X such that every edge ($\mathcal{C}$- or $\mathcal{D}$-) induces a connected subgraph of this cycle. We propose an algorithm to color the $(r,r)$-uniform, complete, circular, mixed hypergraphs for all feasible values with no gaps. In doing so, we show $\chi \left(\mathcal{H}\right)=2$ and $\overline{\chi }$$\left(\mathcal{H}\right)=n-s$ or $n-s-1$ where s is the sieve number.

The bichromatic triangle polynomial $P_G\left(k\right)$ counts vertex k-colorings in which every triangle uses exactly two colors. We develop a transfer matrix framework for three canonical families of 2-trees: book graphs $B_n$, 1-fans $F_n^1$, and triangulated ladders $T{L}_m$. In each case, $P_G\left(k\right)$ satisfies a second-order linear recurrence with an explicit closed form; for $T{L}_m$ this yields a Chebyshev representation, while for $F_n^1$ the binary specialization gives $P_{F_n^1}\left(2\right)=2F_{n+1}$. A spectral identity ${\alpha }^2=r_{+}$ links the dominant characteristic roots of the fan and ladder recurrences, implying identical exponential growth rates when indexed by vertex count, whereas book graphs grow strictly faster for $k\ge 4$. In fact, this correspondence is exact: for all $k\ge 2$, the triangulated ladder polynomial coincides with that of a suitably indexed 1-fan. Passing to line graphs, we interpret $P_{L\left(K_n\right)}\left(k\right)$ as counting edge colorings of $K_n$ that forbid both monochromatic and rainbow triangles, and we identify a sharp obstruction threshold at $n\ge 6$.

Let $S$ be a finite set of positive integers. A mixed hypergraph ${\cal H}$ is a one-realization of $S$ if its feasible set is $S$ and each entry of its chromatic spectrum is either 0 or 1. In [P. Zhao, K. Diao, Y. Chang and K. Wang, The smallest one-realization of a given set \uppercase\expandafter{\romannumeral2}, Discrete Math. 312 (2012) 2946--2951], we determined the minimum number of vertices of a 3-uniform bi-hypergraph which is a one-realization of $S$. In this paper, we generalize this result to $r$-uniform bi-hypergraphs.