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Po-Shen Loh

Publications and source records attributed to Po-Shen Loh.

38 records · Page 3Linked to original sources
arXiv2007

On the strong chromatic number of random graphs

Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colorable if for every partition of V(G) into disjoint sets V_1 \cup ... \cup V_r, all of size exactly k, there exists a proper vertex k-coloring of G with each color appearing exactly once in each V_i. In the case when k does not divide n, G is defined to be strongly k-colorable if the graph obtained by adding k \lceil n/k \rceil - n isolated vertices is strongly k-colorable. The strong chromatic number of G is the minimum k for which G is strongly k-colorable. In this paper, we study the behavior of this parameter for the random graph G(n, p). In the dense case when p >> n^{-1/3}, we prove that the strong chromatic number is a.s. concentrated on one value Δ+1, where Δis the maximum degree of the graph. We also obtain several weaker results for sparse random graphs.

math.CO↗
arXiv2007

Independent transversals in locally sparse graphs

Let G be a graph with maximum degree Δwhose vertex set is partitioned into parts V(G) = V_1 \cup ... \cup V_r. A transversal is a subset of V(G) containing exactly one vertex from each part V_i. If it is also an independent set, then we call it an independent transversal. The local degree of G is the maximum number of neighbors of a vertex v in a part V_i, taken over all choices of V_i and v \not \in V_i. We prove that for every fixed ε> 0, if all part sizes |V_i| >= (1+ε)Δand the local degree of G is o(Δ), then G has an independent transversal for sufficiently large Δ. This extends several previous results and settles (in a stronger form) a conjecture of Aharoni and Holzman. We then generalize this result to transversals that induce no cliques of size s. (Note that independent transversals correspond to s=2.) In that context, we prove that parts of size |V_i| >= (1+ε)[Δ/(s-1)] and local degree o(Δ) guarantee the existence of such a transversal, and we provide a construction that shows this is asymptotically tight.

math.CO↗