Edge colorings of graphs on surfaces and star edge colorings of sparse graphs

In my dissertation, I present results on two types of edge coloring problems for graphs. For each surface Σ, we define ∆(Σ) = max{∆(G)| G is a class two graph with maximum degree ∆(G) that can be embedded in Σ}. Hence Vizing’s Planar Graph Conjecture can be restated as ∆(Σ) = 5 if Σ is a sphere. For a surface Σ with characteristic χ(Σ) ≤ 0, it is known ∆(Σ) ≥ H(χ(Σ))−1, where H(χ(Σ)) is the Heawood number of the surface, and if the Euler char- acteristic χ(Σ) ∈ {−7, −6, . . . , −1, 0}, ∆(Σ) is already known. I study critical graphs on general surfaces and show that (1) if G is a critical graph embeddable on a surface Σ with Euler character- istic χ(Σ) ∈ {−6, −7}, then ∆(Σ) = 10, and (2) if G is a critical graph embeddable on a surface Σ with Euler characteristic χ(Σ) ≤ −8, then ∆(G) ≤ H(χ(Σ)) (or H(χ(Σ))+1) for some special families of graphs, namely if the minimum degree is at most 11 or if ∆ is very large et al. As applications, we show that ∆(Σ) ≤ H (χ(Σ)) if χ(Σ) ∈ {−22, −21, −20, −18, −17, −15, . . . , −8}and ∆(Σ) ≤ H (χ(Σ)) + 1 if χ(Σ) ∈ {−53, . . . , 23, −19, −16}. Combining this with [19], it follows that if χ(Σ) = −12 and Σ is orientable, then ∆(Σ) = H(χ(Σ)). A star k-edge-coloring is a proper k-edge-coloring such that every connected bicolored sub- graph is a path of length at most 3. The star chromatic index χ′st(G) of a graph G is the smallest integer k such that G has a star k-edge-coloring. The list star chromatic index ch′st(G) is defined analogously. Bezegova et al. and Deng et al. independently proved that χ′ (T) ≤ 3∆ for anyst 2 tree T with maximum degree ∆. Here, we study the list star edge coloring and give tree-like bounds for (list) star chromatic index of sparse graphs. We show that if mad(G) \u3c 2.4, then χ′ (G)≤3∆+2andifmad(G)\u3c15,thench′ (G)≤3∆+1.Wealsoshowthatforeveryε\u3e0st 2 7 st 2 there exists a constant c(ε) such that if mad(G) \u3c 8 − ε, then ch′ (G) ≤ 3∆ + c(ε). We also3 st 2 find guaranteed substructures of graph with mad(G) \u3c 3∆ − ε which may be of interest in other2 problems for sparse graphs

Application of the Denovo Discrete Ordinates Radiation Transport Code to Large-Scale Fusion Neutronics

Fusion energy systems pose unique challenges to the modeling and simulation community. These challenges must be met to ensure the success of the ITER experimental fusion reactor. ITER's complex systems require detailed modeling that goes beyond the scale of comparable simulations to date. In this work, the Denovo radiation transport code was used to calculate neutron fluence and kerma for the JET streaming benchmark. This work was performed on the Titan supercomputer at the Oak Ridge Leadership Computing Facility. Denovo is a novel three-dimensional discrete ordinates transport code designed to be highly scalable. Sensitivity studies have been completed to examine the impact of several deterministic parameters. Results were compared against experiment as well as the MCNP and Shift Monte Carlo codes