Daqing Yang

Orderings on graphs and game coloring number

Many graph theoretic algorithms rely on an initial ordering of the vertices of the graph which has some special properties. We discuss new ways to measure the quality of such orders, give methods for constructing high quality orders, and provide applications for these orders. While our main motivation is the study of game chromatic number, there have been other applications of these ideas and we expect there will be more.

Very Asymmetric Marking Games

We investigate a competitive version of the coloring number of a graph G = (V, E). For a fixed linear ordering L of V let s (L) be one more than the maximum outdegree of G when G is oriented so that x ← y if x <Ly. The coloring number of G is the minimum of s (L) over all such orderings. The (a, b)-marking game is played on a graph G = (V, E) as follows. At the start all vertices are unmarked. The players, Alice and Bob, take turns playing. A play consists of Alice marking a unmarked vertices or Bob marking b unmarked vertices. The game ends when there are no remaining unmarked vertices. Together the players create a linear ordering L of V defined by x < y if x is marked before y. The score of the game is s (L). The (a, b)-game coloring number of G is the minimum score that Alice can obtain regardless of Bob’s strategy. The usual (1, 1)-marking game is well studied and there are many interesting results. Our main result is that if G has an orientation with maximum outdegree k then the (k, 1)-game coloring number of G is at most 2k + 2. This extends a fundamental result on the (1, 1)-game coloring number of trees. We also construct examples to show that this bound is tight for many classes of graphs. Finally we prove bounds on the (a, 1)-game coloring number when a < k.

List Total Weighting of Graphs

A graph G=(V, E) is (k, k′)-total weight choosable if the following is true: For any (k, k′)-total list assignment L that assigns to each vertex v a set L(v) of k real numbers as permissible weights, and assigns to each edge e a set L(e) of k′ real numbers as permissible weights, there is a proper L-total weighting, i.e., a mapping f:V∪E→ℝ such that f(y)∈L(y) for each y∈V∪E, and for any two adjacent vertices u and v, ∑ e∈E(u) f(e)+f(u)≠∑ e∈E(v) f(e)+f(v). This Paper introduces a method, the max-min weighting method, for finding proper L-total weightings of graphs. Using this method, we prove that complete multipartite graphs of the form K n,m,1,1,...,1 are (2,2)-total weight choosable and complete bipartite graphs other than K 2 are (1,2)-total weight choosable.

Round robin with look ahead: a new scheduling algorithm for Bluetooth

We propose two new media access control (MAC) scheduling algorithms for Bluetooth whose objective is to achieve high channel utilization (throughput). Conventional scheduling policies such as round robin (RR) in the Bluetooth environment results in wastage of slots and hence poor utilization of the network resources. As Bluetooth devices are designed to carry both voice and data, scheduling becomes a complex task as slots are reserved for voice traffic at periodic intervals and the data packets are allowed to have variable size. We view the MAC scheduling problem in Bluetooth as an online bin packing problem. The two scheduling policies being proposed in the paper, look ahead (LA) and look ahead round robin (LARR), can be viewed as online bin packing with lookahead. We first analytically demonstrate that an optimal scheduling policy can have about 66% improvement in throughput over the round robin policy. Our extensive simulation shows that both LA and LARR achieves nearly 10% improvement in throughput over RR. As the computational complexity of LARR is lower than that of LA and also LARR avoids the possibility of starvation, we suggest the use of this algorithm over RR.

The Two-Coloring Number and Degenerate Colorings of Planar Graphs

The two-coloring number of graphs, which was originally introduced in the study of the game chromatic number, also gives an upper bound on the degenerate chromatic number as introduced by Borodin. It is proved that the two-coloring number of any planar graph is at most nine. As a consequence, the degenerate list chromatic number of any planar graph is at most nine. It is also shown that the degenerate diagonal chromatic number is at most 11 and the degenerate diagonal list chromatic number is at most 12 for all planar graphs.