An Unsolved Graph Theory Problem: Comparing Solutions of Grades 4, 6, & 8

This study investigated how students in Grades 4, 6, and 8 reasoned through a non-routine, unsolved problem. The study took place at a K-8 school in the Midwestern United States. Each grade participated in two or three task-based sessions lasting between 45 and 60 minutes with the researchers. During the sessions, students engaged in the Graceful Tree Conjecture where they examined graceful labelling for Star, Path, and Caterpillar Graphs. We examined differences in students’ generalized solutions across the grades and how they were able to provide justifications and state generalizations of a graceful labelling for the graphs in the Path Class. Descriptions of students’ generalized solutions are included for each grade level.

A Study on Radio Labeling of Diameter N-2 and Caterpillar Graphs

Radio labeling of graphs is a specific type of graph labeling. The basic type of graph labeling is vertex coloring; this is where the vertices of a graph G are assigned different colors so that adjacent vertices are not given the same color. A k-coloring of a graph G is a coloring that uses k colors. The chromatic number of a graph G is the minimum value for k such that a k-coloring exists for G [2].