In 1998 the second author proved that there is an ǫ > 0 such that every graph satisfies χ ≤ ⌈(1 − ǫ)(∆ + 1) + ǫω⌉. The first author recently proved that any graph satisfying ω > 2 3 (∆+1) contains a stable set intersecting every maximum clique. In this note we exploit the latter result to give a much shorter, simpler proof of the former. We include, as a certificate of simplicity, an appendix t...

We prove that for k ≥ 3, the bound given by Brooks’ theorem on the chromatic number of k-th powers of graphs of maximum degree ∆ ≥ 3 can be lowered by 1, even in the case of online list coloring.

We give an inequality for the group chromatic number of a graph as an extension of Brooks’ Theorem. Moreover, we obtain a structural theorem for graphs satisfying the equality and discuss applications of the theorem. Keywords— Brooks’ Theorem, Group coloring, Group chromatic number.

Wilf’s eigenvalue upper bound on the chromatic number is extended to the setting of digraphs. The proof uses a generalization of Brooks’ Theorem to digraph colorings.

The Tachytrechus alatus species group is revised. The species group is defined within the genus Tachytrechus on the basis of possession of a strong basiventral bristle on the fore femur and very reduced pulvilli on the midand hindlegs of both sexes. Males also possess a strongly reduced vertical bristle, and a cluster of 2–3 strong bristles on both the fore femur and mid coxa. The T. alatus spe...

Brooks’ Theorem [R. L. Brooks, On Colouring the Nodes of a Network, Proc. Cambridge Philos. Soc. 37:194-197, 1941] states that every graph G with maximum degree ∆, has a vertex-colouring with ∆ colours, unless G is a complete graph or an odd cycle, in which case ∆ + 1 colours are required. Lovász [L. Lovász, Three short proofs in graph theory, J. Combin. Theory Ser. 19:269-271, 1975] gives an a...

The Neotropical dolichopodine genus Cheiromyia is revised. The limits of the genus are redefined to include only those species with one or more elongate projections on the male postpedicel. Cheiromyia maculipennis (Van Duzee) is transferred to Paraclius Loew comb. nov. Cheiromyia is now known from Costa Rica south to Bolivia and the Brazilian Atlantic Forest, and includes the following five spe...

A vertex-coloring of a hypergraph is conflict-free, if each edge contains a vertex whose color is not repeated on any other vertex of that edge. Let f(r,∆) be the smallest integer k such that each r-uniform hypergraph of maximum vertex degree ∆ has a conflict-free coloring with at most k colors. As shown by Tardos and Pach, similarly to a classical Brooks’ type theorem for hypergraphs, f(r,∆) ≤...

Chen, Lih, and Wu conjectured that for r ≥ 3, the only connected graphs with maximum degree at most r that are not equitably r-colorable are Kr,r (for odd r) and Kr+1. If true, this would be a joint strengthening of the Hajnal-Szemerédi Theorem and Brooks' Theorem. Chen, Lih, and Wu proved that their conjecture holds for r = 3. In this paper we study properties of the hypothetical minimum count...

On riverine flood plains, reorganization by fluvial processes creates and maintains a mosaic of aquatic and riparian landscape elements across a biophysical gradient of disturbance and succession. Across flood plains of gravel-bottom rivers, spring brooks emerge from points of groundwater discharge that may occur in distinct landscape positions. We investigated how ecosystem processes in spring...