On $t$-Common List-Colorings

In this paper, we introduce a new variation of list-colorings. For a graph G and for a given nonnegative integer t, a t-common list assignment of G is a mapping L which assigns each vertex v a set L(v) of colors such that given set of t colors belong to L(v) for every v ∈ V (G). The t-common list chromatic number of G denoted by cht(G) is defined as the minimum positive integer k such that there exists an L-coloring of G for every t-common list assignment L of G, satisfying |L(v)| > k for every vertex v ∈ V (G). We show that for all positive integers k, ` with 2 6 k 6 ` and for any positive integers i1, i2, . . . , ik−2 with k 6 ik−2 6 · · · 6 i1 6 `, there exists a graph G such that χ(G) = k, ch(G) = ` and cht(G) = it for every t = 1, . . . , k − 2. Moreover, we consider the t-common list chromatic number of planar graphs. From the four color theorem [1, 2] and the result of Thomassen [9], for any t = 1 or 2, the sharp upper bound of t-common list chromatic number of planar graphs is 4 or 5. Our first step on t-common list chromatic number of planar graphs is to find such a sharp upper bound. By constructing a planar graph G such that ch1(G) = 5, we show that the sharp upper bound for 1-common list chromatic number of planar graphs is 5. The sharp upper bound of 2-common list chromatic number of planar graphs is still open. We also suggest several questions related to t-common list chromatic number of planar graphs.