# Problem Garden

The problems submitted by GRWC participants in their applications are used to start research collaborations at the workshop. After the collaborations have had time to start and build initial results, *all* problem statements of the participants are made public in this Problem Garden.

Notes:

- A problem with a * at the end corresponds to a paper generated by GRWC group, as listed on our Research Papers page.
- A problem with a ** at the end corresponds to a paper from a GRWC group that is being actively worked. Please contact Combinatorics.Workshop@gmail.com before starting work on these problems.

## 2017 GRWC held in Denver, CO

Heather Smith - Local Dimension for Posets *

Sarah Loeb - Independent Sets in Planar Graphs

Eric Bucher - Permutations of Chip Firings**

Derek Young - Spectral Radius and Distance Spectra

Tomas Masarik - Complexity of Packing Colorings*

Kirk Boyer - Uniform and Preferential Attachment Graphs

Ariel Keller - Degree Sum Conditions to Imply Cycles

Westin King - Parking Functions

Sean English - Sylvester Colorings of Cubic Multigraphs

Michael Tait - Spectral Radius Ramsey Numbers

Jinha Kim - Word Representability of Graphs

Jane Breen - Random Walks and the Normalized Laplacian

Eric Sullivan - Saturation Numbers of Edge-colored Graphs*

Katherine Perry - Rainbow Spanning Trees

Shuliang Bai - Routing on Graphs

Alex Neal Riasanovsky - Cycles of Length Power of 2 in Cubic Graphs

Andrew Uzzell - Positional Games on Posets

Hanlin Zou - Covers of Generalized Quadrangles

Boris Brimkov - Counting the Number of Distinct Zero Forcing Sets*

Juergen Kritschgau - Rainbow Numbers with Respect to Matchings

Ken Duna - Arithmetical Structures on Graphs

Casey Tompkins - Problem on Posets

Zhiyu Wang - Erdos-Szekeres for Cyclic Permutations

Josh Carlson - Throttling for Pursuit-evasion Games on Graphs*

Charles Burnette - Superpositions of Permutations

Da Qi Chen - On Asymmetric Euclidean Ramsey Theory

Joseph Doolittle - Sides of Polytopes

Nathan Graber - Game Saturation Number

Guangming Jing - Sign Patterns with Larger Rational Minimum Rank

Dan Johnston - Bounds on k-Ramsey Numbers

Rachel Kirsch - Universal Partial Words

JD Nir - Lattice Ramsey Numbers

Michael Phillips - Tree and Wheel Ramsey Numbers

Adam Purcilly - Euclidean Ramsey Theory

Carolyn Reinhart - Zero Forcing

Alex Schulte - Anti-van der Waerden

Warren Shull - Hypergraph Coloring Problem Enzo Wendler - Generalization of Skew Adjacency Matrices

Xiaowei Yu - Neighbor-sum Distinguishing Colorings

## 2016 GRWC held in Laramie, WY

- Universal Partial Words*
- Some Open Problems in Spectral Graph Theory*
- Structural Aspects of Bootstrap Percolation*
- Fullerene Problem Proposal**
- A Selection of Graph Partition Problems
- Injective Colorings of Planar Graphs*
- Understanding the Descent Set Polynomial
- 6-critical Graphs with 5 Crossings
- Saturation for Berge-Hypergraphs**
- The Online Happy Ending Problem
- Largest Families of Subsets with Forbidden Subposets**
- d-Matching polynomial**
- Algebraic Connectivity*
- Problems in q-analog t-designs over Galois Fields
- Morphisms of Permutation Graphs**
- Shelling graphs
- Characterizing Chromatic and Flow Polynomials of Graphs
- Acyclic Colorings and Acyclic Edge Colorings
- (p,q)-Colorings*
- Sigma-polynomials

- On-Line Size Ramsey Numbers variant 1
- On-Line Size Ramsey Numbers variant 2
- Random Threshold Hypergraphs
- d-distance Vertex-distinguishing Index
- Detecting Sumsets in Groups
- The Turnpike Problem (Distance Multisets)
- Partitioning into Locally Irregular Subgraphs
- Two Graph Coloring Problems
- Hadwiger Number
- The Regular and Misere Form of Edge-NIM on General Graphs
- The Saturation Number of a Union of Three Cliques of Different Order
- The Earth-Moon Problem
- Probabilistic Consequences of Polynomial Recurrences
- Knot Mosaics and the Mosaic Number
- A Dimension-Reduction Analogue for Integer Programming
- Fractional Ramsey Theory
- Subtractive Magic Total Labeling
- An Extension of the Longest Increasing Path Problem
- Families of Partitions with Restricted Intersections
- The Enhanced Principal Permanent Rank Characteristic Sequence
- Colored Paths in Tournaments

## 2015 GRWC held in Ames, IA

- On the Distance Spectra of Graphs*
- A Problem on {a,b}-factors
- 3-regular Subgraphs of 4-regular Pseudographs*
- Preference Voting
- Subset Divisors
- Counting Cycles in Tournaments
- Finding Pairs of Vertices with the Same Triangle Degree**
- Coupled Choosability with Separation
- Building a Graph One Vertex at a Time*
- Hyperfocused Arcs Revisited
- Hypergraph Versions of a Problem of Erdos and Moser
- Rainbow Matchings in Strongly Edge-Colored Graphs
- Tree Sign Patterns
- Decomposing the Complete Uniform Hypergraph into Hamiltonian l-cycles
- Magic Finite Projective Planes
- Balanced Tableaux
- (q+1)-arcs
- Turan Numbers for Matchings in Complete r-partite Graphs
- 6-crown-free Families
- Uniquely Saturated Graphs for Trees

- Zero-Forcing Numbers of Counterprisms
- Unimodality of the Independence Sequence of a Graph
- On the Turan Number of Forests
- Turing Completeness of Exactly-k Graph Automata
- Number of Cycles in the Graph of Overlapping Permutations
- Maximum Nullity of Loop Graphs Whose Underlying Graph is Outerplanar
- I,F-Partitions of Graphs*
- Improper Colorings
- Hunters and Rabbits*
- Pebbling Kneser Graphs
- Non-transitive Graph Relations
- A Problem of Erdos on Triangle-free Graphs
- Relationships between Pentominoes and Permutations
- Factors of Regular Graphs*
- Embedding Linear Hypergraphs in R^3
- Finding Homeomorphically Irreducible Spanning Trees in Some Classes of Graphs
- A Combinatorial Model for RNA Folding**
- k-fold Sidon Sets
- Total-weight Choosability of Graphs
- Saturation Numbers for Linear Forests
- Eternal Domination on Graphs

## 2014 GRWC held in Denver, CO

- Antimagic-type Labelings*
- A Problem in Incidence Geometry
- Bootstrap Percolation on the Hexagonal Lattice**
- Choosability of Graph Powers
- Choosing Committees in Agreeable Societies
- Color-Blind Index of Graphs*
- Connectivity of Randomly-Oriented Graphs*
- Counting Maximal Chains in Set Families
- Covering Colored Graphs with Colored Cycles
- Extremal Problems using the Absorbing Method
- Forbidden Induced Posets
- Gray Codes for Acyclic Orientations of Graphs

- Gyarfas' Forbidden Subgraph Conjecture
- Halin Graphs
- Hamburger Moment Problem
- Hyperfocused Arcs**
- Increasing Paths in Edge-Ordered Graphs*
- Inverse Principal Perrank Sequences*
- Minimum Rank Problems**
- PSD Minimum Rank of Hamming Graphs
- Potential Ramsey Numbers
- Pseudo-Winners for Digraphs
- Reconstructing Trees from Symmetric Chromatic Polynomials
- Strong Chromatic Index*
- Turan Numbers for Small Bipartite Graphs