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Associated to any hypergraph is a toric ideal encoding the algebraic relations among its edges. We study these ideals and the combinatorics of their minimal generators, and derive general degree bounds for both uniform and non-uniform…

Given a graph G, the graph associahedron is a simple convex polytope whose face poset is based on the connected subgraphs of G. With the additional assignment of a color palette, we define the colorful graph associahedron, show it to be a…

This paper studies the co-maximal graph $\Om(R)$, the induced subgraph $\G(R)$ of $\Om(R)$ whose vertex set is $R\setminus (U(R)\cup J(R))$ and a retract $\G_r(R)$ of $\G(R)$, where $R$ is a commutative ring. We show that the core of…

Coloring a graph $G$ consists in finding an assignment of colors $c: V(G)\to\{1,\ldots,p\}$ such that any pair of adjacent vertices receives different colors. The minimum integer $p$ such that a coloring exists is called the chromatic…

A proof of the Generalized Road Coloring Problem, independent of the recent work by Beal and Perrin, is presented, using both semigroup methods and Trakhtman's algorithm. Algebraic properties of periodic, strongly connected digraphs are…

A vertex coloring of a given simple graph $G=(V,E)$ with $k$ colors ($k$-coloring) is a map from its vertex set to the set of integers $\{1,2,3,\dots, k\}$. A coloring is called perfect if the multiset of colors appearing on the neighbours…

We propose and investigate a unifying class of sparse random graph models, based on a hidden coloring of edge-vertex incidences, extending an existing approach, Random graphs with a given degree distribution, in a way that admits a…

The paper investigates connections between abstract polytopes and properly edge colored graphs. Given any finite n-edge-colored n-regular graph G, we associate to G a simple abstract polytope P_G of rank n, called the colorful polytope of…

A vertex colouring $f:V(G)\to C$ of a graph $G$ is complete if for any $c_1,c_2\in C$ with $c_1\ne c_2$ there are in $G$ adjacent vertices $v_1,v_2$ such that $f(v_1)=c_1$ and $f(v_2)=c_2$. The achromatic number of $G$ is the maximum number…

Associated to any vector configuration A is a toric ideal encoded by vectors in the kernel of A. Each toric ideal has two special generating sets: the universal Gr\"obner basis and the Graver basis. While the former is generally a proper…

In this paper we define and study a ring associated to a graph that we call the cographic toric face ring, or simply the cographic ring. The cographic ring is the toric face ring defined by the following equivalent combinatorial structures…

For a given number of colors, $s$, the guessing number of a graph is the (base $s$) logarithm of the cardinality of the largest family of colorings of the vertex set of the graph such that the color of each vertex can be determined from the…

For integers $k, r > 0$, a conditional $(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of degree $d(v)$ in $G$ is adjacent to at least $\min\{r, d(v)\}$ differently colored…

A colored graph is a directed graph in which nodes or edges have been assigned colors that are not necessarily unique. Observability problems in such graphs consider whether an agent observing the colors of edges or nodes traversed on a…

A graph-theoretic method, simpler than existing ones, is used to characterize the minimal set of monomial generators for the integral closure of any algebra of polynomials generated by quadratic monomials. The toric ideal of relations…

A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induce a union of disjoint paths. In this paper, we prove that for every connected graph with maximum degree at most three and…

A path in an edge-colored graph is called a monochromatic path if all edges of the path have a same color. We call $k$ paths $P_1,\cdots,P_k$ rainbow monochromatic paths if every $P_i$ is monochromatic and for any two $i\neq j$, $P_i$ and…

A colouring of a graph is "nonrepetitive" if for every path of even order, the sequence of colours on the first half of the path is different from the sequence of colours on the second half. We show that planar graphs have nonrepetitive…

In this paper, perfect k-orthogonal colourings of tensor graphs are studied. First, the problem of determining if a given graph has a perfect 2-orthogonal colouring is reformulated as a tensor subgraph problem. Then, it is shown that if two…

Conditional independence models in the Gaussian case are algebraic varieties in the cone of positive definite covariance matrices. We study these varieties in the case of Bayesian networks, with a view towards generalizing the recursive…