On choosability with separation of planar graphs with lists of different sizes

A (k,d)(k,d)-list assignment LL of a graph GG is a mapping that assigns to each vertex vv a list L(v)L(v) of at least kk colors and for any adjacent pair xyxy, the lists L(x)L(x) and L(y)L(y) share at most dd colors. A graph GG is (k,d)(k,d)-choosable if there exists an LL-coloring of GG for every (k,d)(k,d)-list assignment LL. This concept is also known as choosability with separation. It is known that planar graphs are (4, 1)-choosable but it is not known if planar graphs are (3, 1)-choosable. We strengthen the result that planar graphs are (4, 1)-choosable by allowing an independent set of vertices to have lists of size 3 instead of 4. Our strengthening is motivated by the observation that in (4, 1)-list assignment, vertices of an edge have together at least 7 colors, while in (3, 1)-list assignment, they have only at least 5. Our setting gives at least 6 colors

(4, 2)-Choosability of Planar Graphs with Forbidden Structures

All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constraining the structure of the graph, for any ℓ ∈ {3, 4, 5, 6, 7}, a planar graph is 4-choosable if it is ℓ-cycle-free. In terms of constraining the list assignment, one refinement of k-choosability is choosability with separation. A graph is (k, s)-choosable if the graph is colorable from lists of size k where adjacent vertices have at most s common colors in their lists. Every planar graph is (4, 1)-choosable, but there exist planar graphs that are not (4, 3)-choosable. It is an open question whether planar graphs are always (4, 2)-choosable. A chorded ℓ-cycle is an ℓ-cycle with one additional edge. We demonstrate for each ℓ ∈ {5, 6, 7} that a planar graph is (4, 2)-choosable if it does not contain chorded ℓ-cycles

On Choosability with Separation of Planar Graphs with Forbidden Cycles

We study choosability with separation which is a constrained version of list coloring of graphs. A (k; d)-list assignment L of a graph G is a function that assigns to each vertex v a list L(v) of at least k colors and for any adjacent pair xy, the lists L(x) and L(y) share at most d colors. A graph G is (k; d)-choosable if there exists an L-coloring of G for every (k; d)-list assignment L. This concept is also known as choosability with separation. We prove that planar graphs without 4-cycles are (3; 1)-choosable and that planar graphs without 5-cycles and 6-cycles are (3; 1)-choosable. In addition, we give an alternative and slightly stronger proof that triangle-free planar graphs are (3; 1)-choosable.This is the peer-reviewed version of the following article: Choi, Ilkyoo, Bernard Lidický, and Derrick Stolee. "On choosability with separation of planar graphs with forbidden cycles." Journal of Graph Theory 81, no. 3 (2016): 283-306, which has been published in final form at doi: 10.1002/jgt.21875. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. Posted with permission.</p

On Choosability with Separation of Planar Graphs with Forbidden Cycles

We study choosability with separation which is a constrained version of list coloring of graphs. A (k, d)-list assignment L of a graph G is a function that assigns to each vertex v a list L(v) of at least k colors and for any adjacent pair xy, the lists L(x) and L(y) share at most d colors. A graph G is (k, d)-choosable if there exists an L-coloring of G for every (k, d)-list assignment L. This concept is also known as choosability with separation. We prove that planar graphs without 4-cycles are (3, 1)-choosable and that planar graphs without 5-cycles and 6-cycles are (3, 1)-choosable. In addition, we give an alternative and slightly stronger proof that triangle-free planar graphs are (3, 1)-choosable.

Effectively mapping linguistic abstractions for message-passing concurrency to threads on the Java virtual machine

Efficient mapping of message passing concurrency (MPC) abstractions to Java Virtual Machine (JVM) threads is critical for performance, scalability, and CPU utilization; but tedious and time consuming to perform manually. In general, this mapping cannot be found in polynomial time, but we show that by exploiting the local characteristics of MPC abstractions and their communication patterns this mapping can be determined effectively. We describe our MPC abstraction to thread mapping technique, its realization in two frame- works (Panini and Akka), and its rigorous evaluation using several benchmarks from representative MPC frameworks. We also compare our technique against four default mapping techniques: thread-all, round-robin-task-all, random-task-all and work-stealing. Our evaluation shows that our mapping technique can improve the performance by 30%-60% over default mapping techniques. These improvements are due to a number of challenges addressed by our technique namely: i) balancing the computations across JVM threads, ii) reducing the communication overheads, iii) utilizing information about cache locality, and iv) mapping MPC abstractions to threads in a way that reduces the contention between JVM threads.This is a manuscript of a proceeding published as Choi, Ilkyoo, Bernard Lidický, and Derrick Stolee. "On choosability with separation of planar graphs with forbidden cycles." Journal of Graph Theory 81, no. 3 (2016): 283-306. 10.1145/2858965.2814289. Posted with permission.</p