# 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.

**Higher powers of higher nerves****Fibonomial identities****Preferential attachment with 2D choice****Edge decompositions of complete bipartite graphs**********List coloring square graphs****Minimum rank for matrices of double branched graphs****Piercing numbers in approval voting****Exploring coefficients of minimal polynomials****The Nikiforov technique****Packing coloring and T-coloring of subcubic graphs****A generalization of Riordan arrays****Tetrahedral and trapezoidal posets****Edge maximal embeddings of graphs into surfaces****Hadamard-like matrices****Strategies in the game of Cops and Robbers*********Graph information ratios****Inducibility of cycles****Partitionability Problems****Quasirandom permutation forcing sets****Non-negativity of Elser's invariant**********Invariants of quotients of numerical semigroups*********Triangle-edge decompositions of graphs**********Graphs with circular coloring less than 3****Chip firing with hypergraphs**

**Topology of matching complexes****Power domination in reconfigurable graphs****Cyclic sieving of matchings****Problems related to the Erdos-Gyarfas Conjecture****Bounds on generalised radius for special classes of graphs****Constructions of Cospectral Distance Laplacian Graphs*********On the Hall ratio*********Saturation in the hypercube****Colored Eulerian polynomials and unimodality****Permutation packing density****Minkowski sum of all diagonals of hypercubes****The exponents of k-primitive digraphs****Problems related to parking functions*********Maxmizing H-colorings of graphs with minimum degree****Some problems related to sets of line segments*********Matching sequencibility on graphs****Minimally non-linear patterns****Graceful labeling problems*********Permutation class preimages under stack-sorting algorithm*********Face numbers of self-dual 4-dimensional polytopes****The inertia bound for Hermitian matrices****Comparing Ramsey numbers of ordered graphs*********Critical ideals of graphs and applications********

- Local Dimension for Posets *
- Independent Sets in Planar Graphs
- Permutations of Chip Firings**
- Spectral Radius and Distance Spectra
- Complexity of Packing Colorings*
- Uniform and Preferential Attachment Graphs
- Degree Sum Conditions to Imply Cycles
- Parking Functions
- Sylvester Colorings of Cubic Multigraphs
- Spectral Radius Ramsey Numbers
- Word Representability of Graphs
- Random Walks and the Normalized Laplacian
- Saturation Numbers of Edge-colored Graphs*
- Rainbow Spanning Trees
- Routing on Graphs
- Cycles of Length Power of 2 in Cubic Graphs
- Positional Games on Posets
- Covers of Generalized Quadrangles
- Counting the Number of Distinct Zero Forcing Sets*
- Rainbow Numbers with Respect to Matchings

- Erdos-Szekeres for Cyclic Permutations
- Throttling for Pursuit-evasion Games on Graphs*
- Superpositions of Permutations
- On Asymmetric Euclidean Ramsey Theory
- Sides of Polytopes
- Game Saturation Number
- Sign Patterns with Larger Rational Minimum Rank
- Bounds on k-Ramsey Numbers
- Universal Partial Words
- Lattice Ramsey Numbers
- Tree and Wheel Ramsey Numbers
- Euclidean Ramsey Theory
- Zero Forcing
- Anti-van der Waerden
- Hypergraph Coloring Problem
- Generalization of Skew Adjacency Matrices
- Neighbor-sum Distinguishing Colorings
- Arithmetical Structures on Graphs
- Problem on Posets

- 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

- 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

- 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