New Book: Fundamentals of Graph Theory

I have written a new textbook in graph theory, published by the American Mathematical Society.  It is now available for purchase:

Fundamentals of Graph Theory-AMS
Fundamentals of Graph Theory-Amazon
Fundamentals of Graph Theory-Google Books

Reviews:  London Math Society (p. 61) MAA

FGTcover

DESCRIPTION:
Graph theory is a fascinating and inviting branch of mathematics. Many problems are easy to state and have natural visual representations, inviting exploration by new students and professional mathematicians. The goal of this textbook is to present the fundamentals of graph theory to a wide range of readers. The book contains many significant recent results in graph theory, presented using up-to-date notation. The author included the shortest, most elegant, most intuitive proofs for modern and classic results while frequently presenting them in new ways. Major topics are introduced with practical applications that motivate their development, and which are illustrated with examples that show how to apply major theorems in practice. This includes the process of finding a brute force solution (case-checking) when an elegant solution is not apparent. With over 1200 exercises, internet resources (e.g., the OEIS for counting problems), helpful appendices, and a detailed guide to different course outlines, this book provides a versatile and convenient tool for the needs of instructors at a large variety of institutions.

ADDITIONAL SECTIONS:
The Pigeonhole Principle
Uniquely Colorable Graphs

Errata for Fundamentals of Graph Theory

Special Homework Problems for Fundamentals of Graph Theory

Teaching Resources

I have taught math classes from precalculus and statistics to the calculus sequence and classes for math majors at several different institutions.  I have written a number of handouts to cover material that is covered poorly or omitted in the textbooks that I have used.  Readers are welcome to use them for noncommercial purposes.

General:
Why Can’t I Use a Calculator?
Why Can’t I Make Up a Missed Quiz?
Why Do I Have to Show My Work?

Algebra/Precalculus:
Common Algebra Errors
Conversion Formulas for Units of Length, Area, Volume, and Weight
Trigonometry Review Worksheet

Calculus I:
Basic Skills Exam Sample
Evaluating Infinite Limits
Using the Limit Definition of the Derivative
Finding Zeros Using Calculus
Growth Rates of Functions
Growth Rates Worksheet
Explanations from Calculus I
Theorems of Calculus I

Calculus II:
Derivative Review Worksheet
Basic Skills Exam Sample
Evaluating Trig Integrals
Choosing the Right Integration Technique
Area Between Curves
Arclength and Integrals
Mixture Problems with Equal Inflow and Outflow
Mixture Problems with Equal Inflow and Outflow (with Newton’s Law of Cooling)
The Integral Test
Growth Rates of Sequences
Stirling’s Approximation for n!
The Wallis Product for Pi/2
Choosing the Right Convergence Test
The Binomial Theorem
Deriving an Explicit Formula for the Fibonacci Numbers (uses generating functions)
Evaluating the Sum of Inverse Squares
Explanations from Calculus II

Vector Calculus:
Basics of Conic Sections
Generalizing Algebra and Calculus with Vectors
Generalizing Derivatives in Vector Calculus
Arclength and Integrals (includes polar)

Differential Equations:
Euler’s Method Worksheet
Euler’s Formula

Linear Algebra:
Deriving an Explicit Formula for the Fibonacci Numbers (uses difference equations)

Number Theory:
Basic Results in Number Theory
Infinite Series and Number Theory
Fermat’s Last Theorem Worksheet

Discrete Math/Graph Theory:
Counting Worksheet
Counting Derangements
Counting Techniques and Identities (with inclusion/exclusion)
Counting Techniques and Identities (with graph theory examples)
The Pigeonhole Principle
Eulerian Graphs Worksheet
Hamiltonian Graphs Worksheet
Graph Coloring Worksheet
What Do We Study in Graph Theory?
Types of Graph Characterizations
Big Theorems of Graph Theory

Statistics:
Calculator Commands for Statistics
Correlation and Causation
Probability Rules Worksheet
Statistics Tables
Summary of Hypothesis Tests
Choosing the Right Hypothesis Test

My Research

I completed my dissertation October 28, 2010 at Western Michigan University.  My adviser is Allen Schwenk.
The k-Cores of a Graph (200 page PDF)

Publications:

  1. Structural Results on Maximal k-Degenerate Graphs, Discussiones Mathematicae Graph Theory, 32 4 (2012) 659-676.
  2. Nordhaus-Gaddum Theorems for k-Decompositions, Congr. Num. 211 (2012) 171-183.
  3. Cores and Shells of Graphs, Mathematica Bohemica. 138 1 (2013) 43-59.
  4. Nordhaus-Gaddum Results for Genus, with Arthur White, Discrete Math, 313 6 (2013) 824-9.
  5. Two Short Proofs on Total Domination, Discuss Math Graph Theory, 33 2 (2013) 457-459.
  6. 2-Tone Coloring of Joins and Products of Graphs, Congr. Num. 217 (2013) 171-190.
  7. Degree Sequences of Monocore Graphs, Discuss Math Graph Theory, 34 3 (2014) 585-592.
  8. Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208
  9. MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
  10. t-Tone Colorings of Graphs, with Benjamin Phillips, Utilitas Math, 106 (2018) 85-102.
  11. Collapsible Graphs, Congr. Num. 231 (2018) 165-172.
  12. Minimum Edge Cuts in Diameter 2 Graphs, with Allen Schwenk, Discuss Math Graph Theory, 39 2 (2019) 605-608.
    Original Article Correction Corrected Article
  13. Independence Number of Maximal Planar Graphs, Congr. Num. 234 (2019) 61-68.
  14. A Short Proof of Brooks’ Theorem for Vertex Arboricity, AKCE J. of Graphs Combin. (2020) 419-421.
  15. Wiener Indices of Maximal k-Degenerate Graphs, with Zhongyuan Che, Graphs and Combinatorics, 37 2 (2021) 581-589.
  16. Maximal k-degenerate Graphs with Diameter 2, International Journal of Mathematical Combinatorics 2 (2021) 68-79.
  17. k-Paths of k-Trees, Springer PROMS 388 (2020) 287-291.
  18. Properties of Sierpinski Triangle Graphs, 2021. Accepted by Springer PROMS
  19. How to Count k-Paths, J. Integer Sequences, 25 (2022) Article 22.5.6 (9 pages).
  20. Nordhaus-Gaddum Theorems for Multifactor Decompositions, 2022. Accepted by Springer PROMS
  21. Plane Triangulations Without Spanning 2-Trees, Australas. J. Combin. 85 1 (2023) 82-91.
  22. Extremal Decompositions for Nordhaus-Gaddum Theorems, Discrete Math, 346 7 (2023), 113392 (9 pages).
  23. Irregularities of Maximal k-degenerate Graphs, with Zhongyuan Che, Discrete Applied Math. 331 (2023) 70-87.
  24. 2-Tone Coloring of Chordal and Outerplanar Graphs, Australas. J. Combin. 87 1 (2023) 182-197.
  25. A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5 (76 pages).
  26. 2-Tone Coloring of Cactus Graphs, Discrete Applied Math, 347 (2024) 175-186.
  27. Zagreb Indices of Maximal k-degenerate Graphs, Australas. J. Combin. 89 1 (2024) 167-178.
  28. Plane Triangulations Without Large 2-Trees, with Gunnar Brinkmann, 2022+.  Submitted.
  29. 2-Tone Coloring of Planar Graphs, 2023+. Submitted.
  30. Comparing Degeneracy and Odd Cycle Bounds for Chromatic Number, 2023+. Submitted.

Expository Math Talks:
The Mathematics of Redistricting
Amazing Arclength
Fun with Pythagorean Triples
MegaMenger Graphs
Graph Classes with Size kn-k(k+1)/2
A New Spin on Cyclic Decompositions
Bounds and Algorithms for Graph Coloring

Research Presentations:
Introduction to the k-Cores of a Graph
k-Degenerate Graphs
Nordhaus-Gaddum Theorems for k-Decompositions
Nordhaus-Gaddum Theorems for k-Decompositions (includes coloring)
2-Tone Coloring and Petersen Covers
New Results on 2-Tone Coloring
2-Tone Coloring of Joins and Products of Graphs
How to Count k-Paths
Maximal Planar Graphs without Large Spanning 2-Trees