J un 1 99 9 Some Concepts in List Coloring

J un 1 99 9 On Uniquely List Colorable Graphs ∗

Let G be a graph with n vertices and suppose that for each vertex v in G, there exists a list of k colors, L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k–list colorable graph. Recently M. Mahdian and E.S. Mahmoodian characterized uniquely 2–list colorable graphs. Here we state some results which will pave the way in character...

k-forested choosability of graphs with bounded maximum average degree

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...

Sum List Coloring 2*n Arrays

A graph is f -choosable if for every collection of lists with list sizes specified by f there is a proper coloring using colors from the lists. The sum choice number is the minimum over all choosable functions f of the sum of the sizes in f . We show that the sum choice number of a 2 × n array (equivalent to list edge coloring K2,n and to list vertex coloring the cartesian product K22Kn) is n2 ...

A note on partial list coloring

Let G be a simple graph with n vertices and list chromatic number χl(G) = χl. Suppose that 0 ≤ t ≤ χl and each vertex of G is assigned a list of t colors. Albertson, Grossman and Haas [1] conjectured that at least tn χl vertices of G can be colored from these lists. In this paper we find some new results in partial list coloring which help us to show that the conjecture is true for at least hal...

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