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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…
We propose a new proof technique that aims to be applied to the same problems as the Lov\'asz Local Lemma or the entropy-compression method. We present this approach in the context of non-repetitive colorings and we use it to improve…
The induced $q$-color size-Ramsey number $\hat{r}_{\text{ind}}(H;q)$ of a graph $H$ is the minimal number of edges a host graph $G$ can have so that every $q$-edge-coloring of $G$ contains a monochromatic copy of $H$ which is an induced…
Graph coloring involves assigning colors to the vertices of a graph such that two vertices linked by an edge receive different colors. Graph coloring problems are general models that are very useful to formulate many relevant applications…
We derive sharp upper and lower bounds on the number of intersection points and closed regions that can occur in sets of line segments with certain structure, in terms of the number of segments. We consider sets of segments whose underlying…
In this paper, we introduce a class of graphs which we call average hereditary graphs. Many graphs that occur in the usual graph theory applications belong to this class of graphs. Many popular types of graphs fall under this class, such as…
We present an approach to reduced-order modelling that builds off recent graph-theoretic work for representation, exploration, and analysis of computed states of physical systems (Banerjee et al., Comp. Meth. App. Mech. Eng., 351, 501-530,…
It has been shown experimentally that a decimation algorithm based on Survey Propagation (SP) equations allows to solve efficiently some combinatorial problems over random graphs. We show that these equations can be derived as sum-product…
Conditional independence, graphical models and sparsity are key notions for parsimonious statistical models and for understanding the structural relationships in the data. The theory of multivariate and spatial extremes describes the risk…
We use an entropy based method to study two graph maximization problems. We upper bound the number of matchings of fixed size $\ell$ in a $d$-regular graph on $N$ vertices. For $\frac{2\ell}{N}$ bounded away from 0 and 1, the logarithm of…
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…
The spectral and localization properties of heterogeneous random graphs are determined by the resolvent distributional equations, which have so far resisted an analytic treatment. We solve analytically the resolvent equations of random…
This work considers new entropy-based proofs of some known, or otherwise refined, combinatorial bounds for bipartite graphs. These include upper bounds on the number of the independent sets, lower bounds on the minimal number of colors in…
Inference and optimization of real-value edge variables in sparse graphs are studied using the Bethe approximation and replica method of statistical physics. Equilibrium states of general energy functions involving a large set of real…
Several authors modelled networks ad hoc by oriented or disoriented graphs, whereby the problem of allowance (allocation) of the frequencies at the level of the network was transformed into coloring problem of nodes in the graph. Graph…
We study the color patterns that, for $n$ sufficiently large, are unavoidable in $2$-colorings of the edges of a complete graph $K_n$ with respect to $\min \{e(R), e(B)\}$, where $e(R)$ and $e(B)$ are the numbers of red and, respectively,…
A $(p,q)$-coloring of a graph $G$ is an edge-coloring of $G$ which assigns at least $q$ colors to each $p$-clique. The problem of determining the minimum number of colors, $f(n,p,q)$, needed to give a $(p,q)$-coloring of the complete graph…
We study an analogue of the Ramsey multiplicity problem for additive structures, in particular establishing the minimum number of monochromatic 3-APs in 3-colorings of $\mathbb{F}_3^n$ as well as obtaining the first non-trivial lower bound…
In this paper, from a theoretical perspective, we study how powerful graph neural networks (GNNs) can be for learning approximation algorithms for combinatorial problems. To this end, we first establish a new class of GNNs that can solve a…
According to Paul Erd\H{o}s [Some notes on Tur\'an's mathematical work, J. Approx. Theory 29 (1980), page 4] it was Paul Tur\'an who "created the area of extremal problems in graph theory". However, without a doubt, Paul Erd\H{o}s…