Graph Coloring and the Immersion Order

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Graph Coloring and the Immersion Order
The relationship between graph coloring and the immersion order is considered. Vertex connectivity, edge connectivity and related issues are explored. These lead to the conjecture that, if G requires at least t colors, then G must have immersed within it Kt, the complete graph on t vertices. Evidence in support of such a proposition is presented. For each fixed value of t, there can be only a finite number of minimal counterexamples. These counterexamples are characterized based on Kempe chains, connectivity, cutsets and degree bounds. It is proved that minimal counterexamples must, if any exist, be both 4-vertex-connected and t-edge-connected. The t = 5 case is examined in additional detail. The historical context and probable difficulty of settling this conjecture, as well as specific hurdles to its final resolution, are also discussed. Key Words Graph Theory and Algorithms, Chromatic Number, Immersion Containment 1 Overview The applications of graph coloring are legion. The us...
Faisal N. Abu-Khzam, Michael A. Langston
Added 06 Jul 2010
Updated 06 Jul 2010
Type Conference
Year 2003
Authors Faisal N. Abu-Khzam, Michael A. Langston
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