Related papers: Acyclic orientations and poly-Bernoulli numbers
An orientation of a graph $G$ is proper if any two adjacent vertices have different indegrees. The proper orientation number $\overrightarrow{\chi}(G)$ of a graph $G$ is the minimum of the maximum indegree, taken over all proper…
In this work we advance the understanding of the fundamental limits of computation for Binary Polynomial Optimization (BPO), which is the problem of maximizing a given polynomial function over all binary points. In our main result we…
Extending the work of Godsil and others, we investigate the notion of the inverse of a graph (specifically, of bipartite graphs with a unique perfect matching). We provide a concise necessary and sufficient condition for the invertibility…
In this paper, we study the homology of the coloring complex and the cyclic coloring complex of a complete $k$-uniform hypergraph. We show that the coloring complex of a complete $k$-uniform hypergraph is shellable, and we determine the…
We study the enumeration of graph orientations under local degree constraints. Given a finite graph $G = (V, E)$ and a family of admissible sets $\{\mathsf P_v \subseteq \mathbb{Z} : v \in V\}$, let $\mathcal N (G; \prod_{v \in V} \mathsf…
We introduce two operators on stable configurations of the sandpile model that provide an algorithmic bijection between recurrent and parking configurations. This bijection preserves their equivalence classes with respect to the sandpile…
The Exact Matching problem asks whether a bipartite graph with edges colored red and blue admits a perfect matching with exactly $t$ red edges. Introduced by Papadimitriou and Yannakakis in 1982, the problem has resisted deterministic…
We use Menger's Theorem and K\"onig's Line Colouring Theorem to show that in any tripartite graph with two complete (bipartite) sides the maximum number of pairwise edge-disjoint triangles equals the minimum number of edges that meet all…
The regular embeddings of complete bipartite graphs $K_{n,n}$ in orientable surfaces are classified and enumerated, and their automorphism groups and combinatorial properties are determined. The method depends on earlier classifications in…
Let $G$ be a bipartite graph and its adjacency matrix $\mathbb A$. If $G$ has a unique perfect matching, then $\mathbb A$ has an inverse $\mathbb A^{-1}$ which is a symmetric integral matrix, and hence the adjacency matrix of a multigraph.…
We study the polyhedral properties of three problems of constructing an optimal complete bipartite subgraph (a biclique) in a bipartite graph. In the first problem we consider a balanced biclique with the same number of vertices in both…
A complete bipartite graph $K_{3,3}$, considered as a planar linkage with joints at the vertices and with rods as edges, in general admits only motions as a whole, i.e., is inflexible. Two types of its paradoxical mobility were found by…
We perform a detailed investigation of Bipartite Field Theories (BFTs), a general class of 4d N=1 gauge theories which are defined by bipartite graphs. This class of theories is considerably expanded by identifying a new way of assigning…
A graph is said to be nearly complete bipartite if it can be obtained by deleting a set of independent edges from a complete bipartite graph. The nonorientable genus of such graphs is known except in a few cases where the sizes of the…
For each positive integer $n$, the Fibonacci-sum graph $G_n$ on vertices $1,2,\ldots,n$ is defined by two vertices forming an edge if and only if they sum to a Fibonacci number. It is known that each $G_n$ is bipartite, and all Hamiltonian…
Let c(G) be the smallest number of edges we have to test in order to determine an unknown acyclic orientation of the given graph G in the worst case. For example, if G is the complete graph on n vertices, then c(G) is the smallest number of…
The lacking polynomial is a graph polynomial introduced by Chan, Marckert, and Selig in 2013 that is closely related to the Tutte polynomial of a graph. It arose by way of a generalization of the Abelian sandpile model and is essentially…
We introduce a new graph polynomial that encodes interesting properties of graphs, for example, the number of matchings and the number of perfect matchings. Most importantly, for bipartite graphs the polynomial encodes the number of…
There are three usual definitions of a maximum bipartite clique (biclique) in a bipartite graph\,: either maximizing the number of vertices, or of edges, or finding a maximum balanced biclique. The first problem can be solved in polynomial…
We show that there are $k$ simple graphs whose Kronecker covers are isomorphic to the bipartite Kneser graph $H(n,k)$, and that their chromatic numbers coincide with $\chi(K(n,k)) = n - 2k + 2$. We also determine the automorphism groups of…