A Graph Theorist Plants a Tree

Summary

A graph theorist, while planting a tree, is inspired to study coloring problems for a class of graphs. Namely, she completely determines the chromatic number for certain Cayley graphs associated to the cross product of the integers with itself finitely many times, modulo a cyclic subgroup. This result she dubs the “Tree Guard Theorem.” She finds examples “in nature” of this theorem in action, and then seeds are sown for new directions to pursue.

Acknowledgments

The authors would like to thank Gretchen Ludders for observing that the object in Figure 1 resembles a tree guard. We also thank the referee and the editor for many helpful suggestions.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Angel Carrillo

Angel Carrillo was a student at California State University, Los Angeles, whereafter he briefly studied mathematics at Northwestern University. He is currently studying actuarial mathematics and pursuing actuarial designation. He enjoys cooking various cuisines and long walks alongside Lake Michigan with his wife.

Jonathan Cervantes

Jonathan Cervantes is a Ph.D. student at the University of California, Riverside. He was born, raised, and resides in beautiful East Los Angeles. He is interested in graph theory for the naive reasons that it lends itself to beautiful pictures and problems that can be posed to a general audience. Beyond his academic endeavors, he is an amateur musician and an avid gamer. He has a particular fondness for World of Warcraft and Super Smash Bros Melee for the Nintendo Gamecube.

Mike Krebs

Mike Krebs ([email protected]) is a professor of mathematics at California State University, Los Angeles. His current research is in graph theory. He enjoys playing ultimate frisbee, writing songs, and constructing crosswords—one life goal is to have a puzzle published in The New York Times. It will happen, someday.

Francisco Leon

Francisco Leon was a student at California State University, Los Angeles.