Equitable Coloring in 1-Planar Graphs
For every $r\ge13$, we show every 1-planar graph $G$ with $\Delta(G)\le r$ has an equitable $r$-coloring.
math.CO↗
arXiv subjects
Daniel Cranston
Publications and source records attributed to Daniel Cranston.
Short Proofs for Cut-and-Paste Sorting of Permutations
We consider the problem of determining the maximum number of moves required to sort a permutation of $[n]$ using cut-and-paste operations, in which a segment is cut out and then pasted into the remaining string, possibly reversed. We give short proofs that every permutation of $[n]$ can be transformed to the identity in at most $\flr{2n/3}$ such moves and that some permutations require at least $\flr{n/2}$ moves.
math.CO↗
A Strong Edge-Coloring of Graphs with Maximum Degree 4 Using 22 Colors
In 1985, Erdős and Neśetril conjectured that the strong edge-coloring number of a graph is bounded above by ${5/4}Δ^2$ when $Δ$ is even and ${1/4}(5Δ^2-2Δ+1)$ when $Δ$ is odd. They gave a simple construction which requires this many colors. The conjecture has been verified for $Δ\leq 3$. For $Δ=4$, the conjectured bound is 20. Previously, the best known upper bound was 23 due to Horak. In this paper we give an algorithm that uses at most 22 colors.
math.CO↗