22 citations · 92 across the 73 of their papers we have counts for
10 papers · 1 filter
Painting Squares in Shades
Daniel W. Cranston, Landon Rabern
Cranston and Kim conjecture that if is a connected graph with maximum degree and is not a Moore Graph, then ; here is the list chromatic numbe…
Choice number of complete multipartite graphs and
Wenjie He, Lingmin Zhang, Daniel W. Cranston +2
A graph is called \emph{chromatic-choosable} if its choice number is equal to its chromatic number, namely . Ohba has conjectured that every graph satisfying $|…
Multigraphs with are Totally--choosable
Daniel W. Cranston
The \emph{total graph} of a multigraph has as its vertices the set of edges and vertices of and has an edge between two vertices if their corresponding elements are…
An Introduction to the Discharging Method via Graph Coloring
Daniel W. Cranston, Douglas B. West
We provide a "how-to" guide to the use and application of the Discharging Method. Our aim is not to exhaustively survey results proved by this technique, but rather to demystify th…
Graphs with have big cliques
Daniel W. Cranston, Landon Rabern
Brooks' Theorem states that if a graph has and , then . Borodin and Kostochka conjectured that if and , then . We show that if…
1,2,3-Conjecture and 1,2-Conjecture for Sparse Graphs
Daniel W. Cranston, Sogol Jahanbekam, Douglas B. West
We apply the Discharging Method to prove the 1,2,3-Conjecture and the 1,2-Conjecture for graphs with maximum average degree less than 8/3. Stronger results on these conjectures hav…
On the Boundedness of Positive Solutions of the Reciprocal Max-Type Difference Equation with Periodic Parameters
Daniel W. Cranston, Candace M. Kent
We investigate the boundedness of positive solutions of the reciprocal max-type difference equation \[ x_{n}=\max\left\{\frac{A_{n-1}^{1}}{x_{n-1}}, \frac{A_{n-1}^{2}}{x_{n-2}}, \l…
Choosability of the square of a planar graph with maximum degree four
Daniel W. Cranston, Rok Erman, Riste Škrekovski
We study squares of planar graphs with the aim to determine their list chromatic number. We present new upper bounds for the square of a planar graph with maximum degree .…
Regular graphs of odd degree are antimagic
Daniel W. Cranston
An antimagic labeling of a graph with edges is a bijection from to such that for all vertices and , the sum of labels on edges incident to…
Sufficient sparseness conditions for G^2 to be (Δ+1)-choosable, when Δ\ge5
Daniel W. Cranston, Riste Škrekovski
We determine the list chromatic number of the square of a graph $\chil(G^2)$ in terms of its maximum degree when its maximum average degree, denoted $\mad(G)$, is sufficiently…