Let G(V,E) be a simple graph with m edges. For a given integer k, a k-shifted antimagic labeling is a bijection f:E(G)→{k+1,k+2,…,k+m} such that all vertices have different vertex-sums, where the vertex-sum of a vertex v is the total of the labels assigned to the edges incident to v. A graph G is {\it k-shifted antimagic} if it admits a k-shifted antimagic labeling. For the special case when k=0, a 0-shifted antimagic labeling is known as {\it antimagic labeling}; and G is {\it antimagic} if it admits an antimagic labeling. A spider is a tree with exactly one vertex of degree greater than two. A spider forest is a graph where each component is a spider. In this article, we prove that certain spider forests are k-shifted antimagic for all k≥0. In addition, we show that for a spider forest G with m edges, there exists a positive integer k0

A k-fold coloring of a graph assigns to each vertex a set of k colors, and color sets assigned to adjacent vertices are disjoint. The kth chromatic number $\chi_k(G)$ of a graph G is the minimum total number of colors needed in a k-fold coloring of G. Given a graph $G=(V,E)$ and an integer $m\geq0$, the m-cone of G, denoted by $\mu_m(G)$, has vertex set $(V\times\{0,1,\dots,m\})\cup\{u\}$ in which u is adjacent to every vertex of $V\times\{m\}$, and $(x,i)(y,j)$ is an edge if $xy\in E$ and $i=j=0$ or $xy\in E$ and $|i-j|=1$. This paper studies the kth chromatic number of the cone graphs. An upper bound for $\chi_k(\mu_m(G))$ in terms of $\chi_k(G)$, k, and m are given. In particular, it is proved that for any graph G, if $m\geq2k$, then $\chi_k(\mu_m(G))\leq\chi_k(G)+1$. We also find a surprising connection between the kth chromatic number of the cone graph of G and the circular chromatic number of G. It is proved that if $\chi_k(G)/k>\chi_c(G)$ and $\chi_k(G)$ is even, then for sufficiently large m, $\chi_k(\mu_m(G))=\chi_k(G)$. In particular, if $\chi(G)>\chi_c(G)$ and $\chi(G)$ is even, then for sufficiently large m, $\chi(\mu_m(G))=\chi(G)$.