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In this paper, a new approach for fuzzy gyronorms on gyrogroups is presented. The relations between fuzzy metrics(in the sense of Morsi), fuzzy gyronorms, gyronorms on gyrogroups are studied. Also, some sufficient conditions, which can make a fuzzy normed gyrogroup to be a topological gyrogroup and a fuzzy topological gyrogroup, are found. Meanwhile, the relations between topological gyrogroups, fuzzy topological gyrogroups and stratified fuzzy topological gyrogroups are studied. Finally, the properties...

The main purpose of this paper is to give a new approach for partial metric spaces. We first provide the new concept of KM-fuzzy partial metric, as an extension of both the partial metric and KM-fuzzy metric. Then its relationship with the KM-fuzzy quasi-metric is established. In particularly, we construct a KM-fuzzy quasi-metric from a KM-fuzzy partial metric. Finally, after defining the notion of partial pseudo-metric systems, a one-to-one correspondence between partial pseudo-metric systems and...

For paths Pₙ, G. Chartrand, L. Nebeský and P. Zhang showed that $a{c}^{\text{'}}\left(Pₙ\right)\le \left(\genfrac{}{}{0pt}{}{n-2}{2}\right)+2$ for every positive integer n, where ac’(Pₙ) denotes the nearly antipodal chromatic number of Pₙ. In this paper we show that $a{c}^{\text{'}}\left(Pₙ\right)\le \left(\genfrac{}{}{0pt}{}{n-2}{2}\right)-n/2-⎣10/n⎦+7$ if n is even positive integer and n ≥ 10, and $a{c}^{\text{'}}\left(Pₙ\right)\le \left(\genfrac{}{}{0pt}{}{n-2}{2}\right)-(n-1)/2-⎣13/n⎦+8$ if n is odd positive integer and n ≥ 13. For all even positive integers n ≥ 10 and all odd positive integers n ≥ 13, these results improve the upper bounds for nearly antipodal chromatic number of Pₙ.

A graph G is said to be chromatic-choosable if ch(G) = χ(G). Ohba has conjectured that every graph G with 2χ(G)+1 or fewer vertices is chromatic-choosable. It is clear that Ohba’s conjecture is true if and only if it is true for complete multipartite graphs. In this paper we show that Ohba’s conjecture is true for complete multipartite graphs $K_{4,3*t,2*(k-2t-2),1*(t+1)}$ for all integers t ≥ 1 and k ≥ 2t+2, that is, $ch\left(K_{4,3*t,2*(k-2t-2),1*(t+1)}\right)=k$, which extends the results $ch\left(K_{4,3,2*(k-4),1*2}\right)=k$ given by Shen et al. (Discrete Math. 308 (2008) 136-143), and $ch\left(K_{4,3*2,2*(k-6),1*3}\right)=k$ given by He...