a proper vertex coloring of a simple graph is $k$-forested if the graph induced by the vertices of any two color classes is a forest with maximum degree less than $k$. a graph is $k$-forested $q$-choosable if for a given list of $q$ colors associated with each vertex $v$, there exists a $k$-forested coloring of $g$ such that each vertex receives a color from its own list. in this paper, we prov...

All graphs in this paper are simple, connected and undirected. We use V (G) and E(G) to denote the set of vertices and the set of edges of a graph G, respectively. And we denote the maximum degree of G by ∆(G). The undefined terminology can be found in [1]. Let G be a graph of order n. For any vertex u ∈ V (G), N(u) denotes the set of all vertices adjacent to vertex u. Obviously, d(u) is equal ...

‎For a coloring $c$ of a graph $G$‎, ‎the edge-difference coloring sum and edge-sum coloring sum with respect to the coloring $c$ are respectively‎ ‎$sum_c D(G)=sum |c(a)-c(b)|$ and $sum_s S(G)=sum (c(a)+c(b))$‎, ‎where the summations are taken over all edges $abin E(G)$‎. ‎The edge-difference chromatic sum‎, ‎denoted by $sum D(G)$‎, ‎and the edge-sum chromatic sum‎, ‎denoted by $sum S(G)$‎, ‎a...

{sl Let $[n]={1,dots, n}$ be colored in $k$ colors. A rainbow AP$(k)$ in $[n]$ is a $k$ term arithmetic progression whose elements have different colors. Conlon, Jungi'{c} and Radoiv{c}i'{c} cite{conlon} prove that there exists an equinumerous 4-coloring of $[4n]$ which is rainbow AP(4) free, when $n$ is even. Based on their construction, we show that such a coloring of $[4n]$...

{sl let $[n]={1,dots, n}$ be colored in $k$ colors. a rainbow ap$(k)$ in $[n]$ is a $k$ term arithmetic progression whose elements have different colors. conlon, jungi'{c} and radoiv{c}i'{c} cite{conlon} prove that there exists an equinumerous 4-coloring of $[4n]$ which is rainbow ap(4) free, when $n$ is even. based on their construction, we show that such a coloring of ...

a modular $k$-coloring, $kge 2,$ of a graph $g$ without isolated vertices is a coloring of the vertices of $g$ with the elements in $mathbb{z}_k$ having the property that for every two adjacent vertices of $g,$ the sums of the colors of the neighbors are different in $mathbb{z}_k.$ the minimum $k$ for which $g$ has a modular $k-$coloring is the modular chromatic number of $g.$ except for some s...

an acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. the acyclic chromatic index of a graph $g$ denoted by $chi_a '(g)$ is the minimum number $k$ such that there is an acyclic edge coloring using $k$ colors. the maximum degree in $g$ denoted by $delta(g)$, is the lower bound for $chi_a '(g)$. $p$-cuts introduced in this paper acts as a powerfu...