We collect some of our favorite proofs of Brooks’ Theorem, highlighting advantages and extensions of each. The proofs illustrate some of the major techniques in graph coloring, such as greedy coloring, Kempe chains, hitting sets, and the Kernel Lemma. We also discuss standard strengthenings of vertex coloring, such as list coloring, online list coloring, and Alon–Tarsi orientations, since analo...

to examine hydraulics of lnapls in soil, the retention curves of diesel fuel and water were both determined by hanging column method. the soil retention parameters were obtained based on van genuchten, brooks-corey and campbell’s models. in addition, the soil hydraulic conductivity for both fluids were predicted based on mualem- brooks-corey, burdine- brooks-corey, mualem-van genuchten and camp...

First, I introduce quantum graph theory. I also discuss a known lower bound on the independence numbers and derive from it an upper bound on the chromatic numbers of quantum graphs. Then, I construct a family of quantum graphs that can be described as tropical, cyclical, and commutative. I also define a step logarithm function and express with it the bounds on quantum graph invariants in closed...

The historical biogeography of freshwater fish helminth parasites is linked to speciation. It is well recognized and documented that bursts of speciation are often limited to certain periods of geological time (Eldredge & Gould 1972; Gould & Eldredge; 1977, Gould, 2002; Hoberg & Brooks, 2008) and is highly dependent on the space in which it occurs (Brooks & McLennan, 2002; Hoberg & Brooks, 2008...

The SDSN-Aggregation: The SDSN aggregation method uses a combination of methods ”Brooks-Iyengar algorithm” and ”Witness based approach for data fusion assurance” to achieve the aggregation. The aggregation method should work for multiple data types and for different densities of the network. The SDSN-aggregation is event-based, it is triggered on an aggregation request by another node or when n...

A graph is called to be uniquely list colorable, if it admits a list assignment which induces a unique list coloring. We study uniquely list colorable graphs with a restriction on the number of colors used. In this way we generalize a theorem which characterizes uniquely 2–list colorable graphs. We introduce the uniquely list chromatic number of a graph and make a conjecture about it which is a...