Electronic Theses and Dissertations

Date of Award


Document Type


Degree Name

Ph.D. in Mathematics



First Advisor

William Staton

Second Advisor

Talmadge James Reid

Third Advisor

Dawn Wilkins

Relational Format



A graph is called well-covered if all of its maximal independent sets have the same cardinality. We give a characterization of well-covered k-trees. A graph is said to be uniquely χ-colorable if, modulo permutations of colors, it has exactly one proper χ-coloring. The k-trees with at least k+1 vertices are minimal uniquely (k +1)-colorable, i.e., they have the minimal number of edges necessary for uniquely (k+1)-colorable graphs. We introduce the k-frames, a new class of minimal uniquely (k+1)-colorable graphs that generalizes the k-trees.

The covering range of a graph is the difference between the cardinality of a largest maximal independent set of a graph and the cardinality of a smallest maximal independent set of the graph. We give the covering range for some cubic graphs and a class of k-regular graphs.

Included in

Mathematics Commons



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