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In 2019, Butler, Choi, Kim, and Seo introduced a new type of juggling card that represents multiplex juggling patterns in a natural bijective way. They conjectured a formula for the generating function for the number of multiplex juggling…
In this article, we use a unified approach to prove several classes of planar graphs are DP-$3$-colorable, which extend the corresponding results on $3$-choosability.
Holroyd and Liggett recently proved the existence of a stationary 1-dependent 4-coloring of the integers, the first stationary k-dependent q-coloring for any k and q. That proof specifies a consistent family of finite-dimensional…
The problem of map enumeration concerns counting connected spatial graphs, with a specified number $j$ of vertices, that can be embedded in a compact surface of genus $g$ in such a way that its complement yields a cellular decomposition of…
It was conjectured by the third author in about 1973 that every $d$-regular planar graph (possibly with parallel edges) can be $d$-edge-coloured, provided that for every odd set $X$ of vertices, there are at least $d$ edges between $X$ and…
We characterize the generating function of the number of representations described in the title in terms of the theory of modular forms. Appealing to this characterization we obtain explicit formulas for the representation numbers as…
A square coloring of a graph $G$ is a coloring of the square $G^2$ of $G$, that is, a coloring of the vertices of $G$ such that any two vertices that are at distance at most $2$ in $G$ receive different colors. We investigate the complexity…
For spin systems, such as the $q$-colorings and independent-set models, approximating the partition function in the so-called non-uniqueness region, where the model exhibits long-range correlations, is typically computationally hard for…
In this paper, we extend the notion of labeled partitions with ordinary permutations to colored permutations in the sense that the colors are endowed with a cyclic structure. We use labeled partitions with colored permutations to derive the…
Despite the fact that some vertex coloring problems are polynomially solvable on certain graph classes, most of these problems are not "under control" from a polyhedral point of view. The equivalence between \emph{optimization} and…
In this paper, we give polynomial-time algorithms that can take a graph G with a given combinatorial embedding on an orientable surface S of genus g and produce a planar drawing of G in R^2, with a bounding face defined by a polygonal…
We study the generating series associated with the degree sequence of a monomial self-map of a projective toric variety. We establish conditions under which this series has its circle of convergence as a natural boundary, and hence is a…
We present exact results on the partition function of the $q$-state Potts model on various families of graphs $G$ in a generalized external magnetic field that favors or disfavors spin values in a subset $I_s = \{1,...,s\}$ of the total set…
Counting problems, determining the number of possible states of a large system under certain constraints, play an important role in many areas of science. They naturally arise for complex disordered systems in physics and chemistry, in…
We give a near-linear time 4-coloring algorithm for planar graphs, improving on the previous quadratic time algorithm by Robertson et al. from 1996. Such an algorithm cannot be achieved by the known proofs of the Four Color Theorem (4CT).…
The problem of sampling edge-colorings of graphs with maximum degree $\Delta$ has received considerable attention and efficient algorithms are available when the number of colors is large enough with respect to $\Delta$. Vizing's theorem…
The chromatic number of an planar graph is not greater than four and this is known by the famous four color theorem and is equal to two when the planar graph is bipartite. When the planar graph is even-triangulated or all cycles are greater…
We study the complexity of counting (weighted) planar graph homomorphism problem $\tt{Pl\text{-}GH}(M)$ parametrized by an arbitrary symmetric non-negative real valued matrix $M$. For matrices with pairwise distinct diagonal values, we…
This paper describes a sequence of natural numbers that grows faster than any Turing computable function. This sequence is generated from a version of the tiling problem, called a coloring system. In our proof that generates the sequence,…
We give Euler-like recursive formulas for the $t$-colored partition function when $t=2$ or $t=3,$ as well as for all $t$-regular partition functions. In particular, we derive an infinite family of ``triangular number" recurrences for the…