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 + d5n/3e. List coloring has been well-studied in recent years (see for example surveys in [1], [7], [9], [11]). The vertices of a graph (often a line graph) are given color lists and we seek to determine if there is a proper coloring using colors from the lists. Typically one seeks the minimum value k such that for every choice of lists of size k there is a proper coloring. In this case we call the graph k-choosable. More generally the list size for each vertex can be specified and one can determine if for every choice of lists of the specified sizes there is a proper coloring. The general problem here would be to characterize which list size assignments allow a proper coloring for every choice of lists of the given sizes. For the case of line graphs of complete bipartite graphs this is called the generalized Dinitz problem in [5] and it is noted that it ‘was proposed independently by (at least) H. Taylor and D. Knuth (personal communications) and the author.’ (The ‘author’ referred to in the previous sentence is Galvin.) The general characterization problem for line graphs of trees (and more generally for vertex coloring graphs with every block a clique) is solved in another paper [8] by the author of this paper. For the k-choosability problem we seek to minimize the maximum list size. Here we will examine another special case of the generalized problem where we seek to minimize the average list size (by minimizing the sum of the list sizes). While there have been a number of generalizations and variations on choosability as well as results with sufficient conditions for choosability based on degrees, it seems that the problem of minimizing the sum of the list sizes has not been examined. ∗Partially supported by a grant from the Reidler Foundation the electronic journal of combinatorics 9 (2002), #N8 1 As noted in the abstract, properly coloring the edges of the complete bipartite graph K2,n corresponds to properly coloring the vertices of the cartesian product K22Kn and also corresponds to properly coloring the cells of a 2×n array with distinct colors in each row and each column. We will adopt the last perspective in this paper. For a collection C = {C(i, j)|i = 1, 2, . . . , m and j = 1, 2, . . . , n} of lists we will say that the m × n array is C-colorable if we can choose colors ci,j ∈ C(i, j) for the cells (i, j) of the array such that the colors selected for each row and column are distinct (that is, ci,j = cr,s ⇒ i 6= r and j 6= s). We will call such a choice a proper C-coloring. For 1 × n arrays we will write C(i) rather than C(1, i). Consider a size function f : {1, 2, . . . , m} × {1, 2, . . . , n} → Z. We will say that the m × n array is f -choosable if for every C with |C(i, j)| = f(i, j), the array is C-colorable. For convenience we allow f(i, j) ≤ 0. If any f(i, j) ≤ 0 then we will say that the array is not f -choosable. We will call ∑m i=1 ∑n j=1 f(i, j) the size of f . Then, the sum choice number of the m × n array is the minimum size of a choosable f . Theorem 1 The sum choice number of the 2 × n array is n + d5n/3e. Proof: This will follow immediately from Lemmas 2 and 3 in Sections 2 and 3. 2. We will use an array notation to represent size functions f . So for example, 2 3 1 2 represents the function f(1, 1) = f(2, 2) = 2, f(1, 2) = 3 and f(2, 1) = 1. We will say that a collection of lists is initial if every list is of the form {1, 2, . . . , s}. List coloring graphs with the restriction that every list is initial is essentially graph coloring. For sum list coloring, if the lists are restricted to be initial, then we get sum coloring (see, for example, [2] for more on this problem). Finding the minimum sum for a coloring is NP-complete even for line graphs of bipartite graphs with maximum degree three [6]. We will refer to greedy coloring with respect to a given ordering of the cells as the coloring obtained by sequentially coloring the cells in the given order, choosing the smallest color at each step that will not violate the conditions for a proper list coloring. Greedy coloring will produce a proper list coloring if, when each cell is colored, the size of its list is greater than the number of previously colored cells in the same row or column having lists that have non-empty intersection with the cell being colored.