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.
","abstract_html":"<p>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. </p> <p>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. </p>","abstract_has_math":false,"creators":["Payne, Wanda Renea"],"institution":null,"degree_name":"Ph.D. in Mathematics","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["William Staton","Talmadge James Reid","Dawn Wilkins"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-01-01T08:00:00Z","date_published":"2013-01-01T08:00:00Z","updated_at":"2026-07-24T03:06:53Z","subjects":["covering range","unique colorability","well-covered","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://egrove.olemiss.edu/etd/1439","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["William Staton","Talmadge James Reid","Dawn Wilkins"]},{"key":"dc:creator","label":"Author","values":["Payne, Wanda Renea"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2020-01-23T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D. in Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["covering range","unique colorability","well-covered","Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://egrove.olemiss.edu/etd/1439"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Well-covered Graphs, Unique Colorability, and Covering Range"]}]}],"canonical_facts":{"dc:contributor":["William Staton","Talmadge James Reid","Dawn Wilkins"],"dc:creator":["Payne, Wanda Renea"],"dc:date.available":["2020-01-23T08:00:00Z"],"dc:description.abstract":["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.
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