5 papers
All--Isolation in Trees
Geoffrey Boyer, Garrett C. Farrell, Wayne Goddard
We define an all--isolating set of a graph to be a set of vertices such that, if one removes and all its neighbors, then no component in what remains has order or mo…
On the isolation number of graphs with minimum degree four
Wayne Goddard, Michael A. Henning
An isolating set in a graph is a set of vertices such that removing and its neighborhood leaves no edge. The isolation number of (also known as the vertex-e…
A Note on Distance-Fall Colorings
Wayne Goddard, Sonwabile Mafunda
We say a proper coloring of a graph is distance- fall if every vertex is within distance of at least one vertex of every color. We show that if is a connected graph of o…
Bounds on Independent Isolation in Graphs
Geoffrey Boyer, Wayne Goddard
An isolating set of a graph is a set of vertices such that, if and its neighborhood is removed, only isolated vertices remain; and the isolation number is the minimum size…
Bounds on Coloring Trees without Rainbow Paths
Wayne Goddard, Tyler Herrman, Simon J. Hughes
For a graph with colored vertices, a rainbow subgraph is one where all vertices have different colors. For graph , let denote the maximum number of different colors in…