arXiv subjects
Publications and source records attributed to Kevin Moss.
Let $G$ be a simple graph and $v$ be a vertex of $G$. The triangle-degree of $v$ in $G$ is the number of triangles that contain $v$. While every graph has at least two vertices with the same degree, there are graphs in which every vertex has a distinct triangle-degree. In this paper, we construct an infinite family of graphs with this property. We also study the vertex degrees and size of graphs with this property.
List coloring generalizes graph coloring by requiring the color of a vertex to be selected from a list of colors specific to that vertex. One refinement of list coloring, called choosability with separation, requires that the intersection of adjacent lists is sufficiently small. We introduce a new refinement, called choosability with union separation, where we require that the union of adjacent lists is sufficiently large. For $t \geq k$, a $(k,t)$-list assignment is a list assignment $L$ where $|L(v)| \geq k$ for all vertices $v$ and $|L(u)\cup L(v)| \geq t$ for all edges $uv$. A graph is $(k,t)$-choosable if there is a proper coloring for every $(k,t)$-list assignment. We explore this concept through examples of graphs that are not $(k,t)$-choosable, demonstrating sparsity conditions that imply a graph is $(k,t)$-choosable, and proving that all planar graphs are $(3,11)$-choosable and $(4,9)$-choosable.
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 $\ell \in \{3,4,5,6,7\}$, a planar graph is 4-choosable if it is $\ell$-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 $\ell$-cycle is an $\ell$-cycle with one additional edge. We demonstrate for each $\ell \in \{5,6,7\}$ that a planar graph is $(4,2)$-choosable if it does not contain chorded $\ell$-cycles.
We study congruences involving truncated hypergeometric series of the form_rF_{r-1}(1/2,...,1/2;1,...,1;λ)_{(mp^s-1)/2} = \sum_{k=0}^{(mp^s-1)/2} ((1/2)_k/k!)^r λ^k where p is a prime and m, s, r are positive integers. These truncated hypergeometric series are related to the arithmetic of a family of algebraic varieties and exhibit Atkin and Swinnerton-Dyer type congruences. In particular, when r=3, they are related to K3 surfaces. For special values of λ, with s=1 and r=3, our congruences are stronger than what can be predicted by the theory of formal groups because of the presence of elliptic curves with complex multiplications. They generalize a conjecture made by Rodriguez-Villegas for the λ=1 case and confirm some other supercongruence conjectures at special values of λ.