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Research Interest

Extremal Graph Theory, Graph Colorings

Articles

Submitted

  1. Brandt, A.; Diemunsch, J.; Erbes, C.; LeGrand, J. and Moffatt, C. A Robber Locating Strategy for Trees.
  2. Brandt, A.; Diemunsch, J. and Jahanbekam, S. Lucky Choice Number of Planar Graphs with Given Girth.
  3. Berikkyzy, Z.; Brandt, A.; Jahanbekam, S.; Larsen, V. and Rorabaugh, D. Antimagic Labelings of Weighted and Oriented Graphs.
  4. Brandt, A.; Irwin, D. and Jiang, T. Stability and Tur'an numbers of a class of hypergraphs via Lagrangians.

Accepted

  1. Brandt, A.; Moran, B.; Nepal, K.; Pfender, F. and Sigler, D. Local Gap Colorings from Edge Labelings. In Australas. J. Combin., Accepted.
  2. Brandt, A.; Ferrara, M.; Kumbhat, M.; Loeb, S.; Stolee, D. and Yancey, M. I,F-Partitions of Sparse Graphs. In European J. Combin., Accepted.


Project Descriptions

A Robber Locating Strategy for Trees [pdf]

A collaboration with fellow CU Denver graduate students, this project developed from a problem presentation session during a weekly graph theory research seminars at CU Denver. As an accessible problem for PhD students without an extensive graph theory background, I presented this problem with the intent of involving early PhD studnets in research. The resulting project was an excellent learning experience in regard to collaborative student research, writing and editing a paper without advisor oversight, and an introduction to the publication process.
Abstract: The robber locating game, introduced by Seager, is a variation of the classic cops and robbers game on a graph. A cop attempts to locate an invisible robber on a graph by probing a single vertex v each turn, from which the cop learns the robber's distance. The robber is then permitted to stay at his current vertex or move to an adjacent vertex other than v. A graph is locatable if the cop is able to locate the robber in a finite number of probes, and the location number of a graph is the minimum number of probes necessary to determine the robber's location regardless of the robber's evasion strategy. We provide a strategy that locates the robber on trees. The number of probes used under this strategy meets, and often improves, the theoretical bound on the location number of trees.

Local Gap Colorings from Edge Labelings [pdf]

A collaboration among students and faculty from a second course in graduate graph theory, this project began during the last week of class and continued into the following academic year. With a talented undergraduate student as part of the group, I developed an appreciation for conducting and promoting research with undergraduate students.
Abstract: We study a local version of gap vertex-distinguishing edge coloring. From an edge labeling of a graph, an induced vertex coloring is obtained by coloring the vertices with the greatest difference between incident edge labels. The local gap chromatic number is the minimum number of edge labels {1,...,k} for which there exists an edge coloring where adjacent vertices receive distinct colors. We prove that k is either the chromatic number or one more for all graphs. Further we find graph classes attaining both bounds.

Lucky Choice Number of Planar Graphs with Given Girth [pdf]

This paper combines two well-known proof techniques: the Combinatorial Nullstellensatz and the Discharging Method. After being introduced to and presenting on the Combinatorial Nullstellensatz in CU Denver's graph theory research seminar and sitting in on a readings course on the Discharging Method, I worked with a fellow graduate student and a postdoc to utilize both of these well-known proof techniques to improve bounds on a particular graph coloring.
Abstract: Suppose the vertices of a graph are labeled with integers {1,...,k}. We seek the minimum k so that summing the labels on the neighborhood of each vertex induces a proper vertex coloring. In 2009, Czerwínski, Grytczuk, and ̇Zelazny conjectured that k is at most the chromatic number of a graph; this conjecture is currently open for bipartite graphs. In this paper, we improve the current bounds for particular classes of planar graphs with a strengthening of the results through a list lucky labeling. We apply the discharging method and the Combinatorial Nullstellensatz to show that for a planar graph of girth at least 26 has minimum k of at most 3. We also show that a graph with girth at least 7,6, and 5 has minimum k at most 8, 9, and 19, respectively.

Antimagic Labelings of Weighted and Oriented Graphs [pdf]

Beginning as a project at the 2014 Graduate Research Workshop in Combinatorics, this project was completed with graduate students from programs across the country and a postdoc and served as a learning opportunity for collaborating at a distance. This paper uses Sage programming to assist in applying the Combinatorial Nullstellensatz to improve known bounds for a graph coloring problem.
Abstract: Suppose the vertices of a graph are weighted with real numbers. We seek the minimum number of options on edges so that an edge labeling can be chosen in such a way that summing a vertex weight with its incident edge labels yields distinct sums. We show that the minimum number of options necessary is at most the number of edges plus 4/3 the number of vertices. In a non-weighted oriented graph, we seek the minimum number of options on edges so that an edge labeling can be chosen in such a way that the sum of labels on edges incident to andoriented toward a vertex minus the sum of labels on edges incident to and oriented away from a vertex yields distinct vertex weights. In this case, we show that the minimum number of options necessary is at most the number of edges plus 2/3 the number of vertices.

I,F-Partitions of Sparse Graphs [pdf]

A project from the 2015 Graduate Research Workshop in Combinatorics, this project was completed with two faculty, one postdoc, one graduate student, and one industry researcher and served as another learning opportunity for collaborating at a distance.
Abstract: A star k-coloring is a proper k-coloring where the union of two color classes induces a star forest. While every planar graph is 4-colorable, not every planar grapn is star 4-colorable. One method to produce star 4-colorings is to partition the vertex set into a 2-independent set and a forest; such a partition is called an I,F-partition. We use a combination of potential functions and the discharging method to prove that every graph with maximum average degree less than 2.5 has an I,F-partition, which is sharp and answers a question of Cranston and West. This result implies that planar graphs of girth at least 10 are star 4-colorable, improving upon bounds of Bu, Cranston, Montassier, Raspaud, and Wang.