$4$-Colouring of Generalized Signed Planar Graphs

Assume $G$ is a graph and $S$ is a set of permutations of positive integers. An $S$-signature of $G$ is a pair $(D, \sigma)$, where $D$ is an orientation of $G$ and $\sigma: E(D) \to S$ is a mapping which assigns to each arc $e=(u,v)$ a permutation $\sigma(e)$ in $S$. We say $G$ is $S$-$k$-colourable if for any $S$-signature $(D, \sigma)$ of $G$, there is a mapping $f: V(G) \to [k]$ such that for each arc $e=(u,v)$ of $G$, $\sigma(e)(f(u)) \ne f(v)$. The concept of $S$-$k$-colourable is a common generalization of many colouring concepts. This paper studies the problem as to which subsets $S$ of $S_4$, every planar graph is $S$-$4$-colourable. We call such a subset $S$ of $S_4$ a good subset. The Four Colour Theorem is equivalent to saying that $S=\{id\}$ is good. It was proved by Jin, Wong and Zhu (arXiv:1811.08584) that a subset $S$ containing $id$ is good if and only if $S=\{id\}$. In this paper, we prove that, up to conjugation, every good subset of $S_4$ not containing $id$ is a subset of $\{(12),(34),(12)(34)\}$.

Wolf-Swarm Colony for Signature Gene Selection Using Weighted Objective Method

Microarray study has a huge impact on the proper detection and classification of cancer, as it analyzes the changes in expression level of genes which are strongly associated with cancer. In this chapter, a new weighted objective wolf-swarm colony optimization (WOWSC) technique is proposed for the selection of significant and informative genes from the cancer dataset. To extract the relevant genes from datasets, WOWSC utilizes four different objective functions in a weighted manner. Experimental analysis shows that the proposed methodology is very efficient in obtaining differential and biologically relevant genes which are effective for the classification of disease. The technique is able to generate a good subset of genes which offers more useful insight to the gene-disease association.