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Two distinct crossings are independent if the end-vertices of the crossed pair of edges are mutually different. If a graph G has a drawing in the plane such that every two crossings are independent, then we call G a plane graph with independent crossings or IC-planar graph for short. A proper total-k-coloring of a graph G is a mapping c : V (G) ∪ E(G) → {1, 2, . . . , k} such that any two adjacent elements in V (G) ∪ E(G) receive different colors. Let Σc(v) denote the sum of the color of a vertex v and the colors of all incident edges of v. A total-k-neighbor sum distinguishing-coloring of G is a total-k-coloring of G such that for each edge uv ∈ E(G), Σc(u) ≠ Σc(v). The least number k needed for such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by χΣ″(G)\chi _\Sigma ^{''} ( G ) . In this paper, it is proved that if G is an IC-planar graph with maximum degree Δ(G), then chΣ″(G)≤max{Δ(G)+3, 17}ch_\Sigma ^{''} ( G ) \le \max \left\{ {\Delta ( G ) + 3,\;17} \right\} , where chΣ″(G)ch_\Sigma ^{''} ( G ) is the neighbor sum distinguishing total choosability of G.

A strong edge-coloring of a graph is a proper edge-coloring where each color class induces a matching. We denote by 𝜒's(G) the strong chromatic index of G which is the smallest integer k such that G can be strongly edge-colored with k colors. It is known that every planar graph G has a strong edge-coloring with at most 4 Δ(G) + 4 colors [R.J. Faudree, A. Gyárfás, R.H. Schelp and Zs. Tuza, The strong chromatic index of graphs, Ars Combin. 29B (1990) 205–211]. In this paper, we show that if G is a planar graph with g ≥ 5, then 𝜒's(G) ≤ 4(G) − 2 when Δ(G) ≥ 6 and 𝜒's(G) ≤ 19 when Δ(G) = 5, where g is the girth of G.

MicroRNAs (miRNAs) play crucial roles in multiple biological processes and human diseases and can be considered as therapeutic targets of small molecules (SMs). Because biological experiments used to verify SM–miRNA associations are time-consuming and expensive, it is urgent to propose new computational models to predict new SM–miRNA associations. Here, we proposed a novel method called Dual-network Collaborative Matrix Factorization (DCMF) for predicting the potential SM–miRNA associations. Firstly, we utilized the Weighted K Nearest Known Neighbors (WKNKN) method to preprocess SM–miRNA association matrix. Then, we constructed matrix factorization model to obtain two feature matrices containing latent features of SM and miRNA, respectively. Finally, the predicted SM–miRNA association score matrix was obtained by calculating the inner product of two feature matrices. The main innovations of this method were that the use of WKNKN method can preprocess the missing values of association matrix and the introduction of dual network can integrate more diverse similarity information into DCMF. For evaluating the validity of DCMF, we implemented four different cross validations (CVs) based on two distinct datasets and two different case studies. Finally, based on dataset 1 (dataset 2), DCMF achieved Area Under receiver operating characteristic Curves (AUC) of 0.9868 (0.8770), 0.9833 (0.8836), 0.8377 (0.7591) and 0.9836 ± 0.0030 (0.8632 ± 0.0042) in global Leave-One-Out Cross Validation (LOOCV), miRNA-fixed local LOOCV, SM-fixed local LOOCV and 5-fold CV, respectively. For case studies, plenty of predicted associations have been confirmed by published experimental literature. Therefore, DCMF is an effective tool to predict potential SM–miRNA associations. [ABSTRACT FROM AUTHOR]