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A locally irregular multigraph is a multigraph whose adjacent vertices have distinct degrees. The locally irregular edge coloring is an edge coloring of a multigraph $G$ such that every color induces a locally irregular submultigraph of…
In the first part, we introduce a notion a degree of edge-colorings of bicubic plane graphs and proves some local formula of the graded number of colorings. In the second part, we give a new proof of a result of Fisk saying that any two…
We extend two well-known results in Ramsey theory from from $K_n$ to arbitrary $n$-chromatic graphs. The first is a note of Erd\H os and Rado stating that in every 2-coloring of the edges of $K_n$ there is a monochromatic tree on $n$…
Drawings of non-planar graphs always result in edge crossings. When there are many edges crossing at small angles, it is often difficult to follow these edges, because of the multiple visual paths resulted from the crossings that slow down…
In this paper we have investigated some old issues concerning four color map problem. We have given a general method for constructing counter-examples to Kempe's proof of the four color theorem and then show that all counterexamples can be…
The Lov\'{a}sz Local Lemma (LLL) is a cornerstone principle in the probabilistic method of combinatorics, and a seminal algorithm of Moser & Tardos (2010) provides an efficient randomized algorithm to implement it. This can be parallelized…
A well-known conjecture of F\"uredi, Kahn, and Seymour (1993) on non-uniform hypergraph matching states that for any hypergraph with edge weights $w$, there exists a matching $M$ such that the inequality $\sum_{e\in M} g(e) w(e) \geq…
We propose a notion of graph convergence that interpolates between the Benjamini--Schramm convergence of bounded degree graphs and the dense graph convergence developed by L\'aszl\'o Lov\'asz and his coauthors. We prove that spectra of…
Lov\'asz gave a short proof of Brooks' theorem by coloring greedily in a good order. We give a different short proof by reducing to the cubic case. Then we show how to extend the result to (online) list coloring via the Kernel Lemma.
In any vertex coloring of a graph some edges have differently colored ends (\emph{good} edges) and some are monochromatic (\emph{bad} edges). In a proper coloring all edges are good. In a \emph{majority coloring} it is enough that for every…
We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of the results of Kahn, Galvin and Tetali, and Zhao showing…
Recent developments in approximate counting have made startling progress in developing fast algorithmic methods for approximating the number of solutions to constraint satisfaction problems (CSPs) with large arities, using connections to…
The Lovasz Local Lemma [EL75] is a powerful tool to prove the existence of combinatorial objects meeting a prescribed collection of criteria. The technique can directly be applied to the satisfiability problem, yielding that a k-CNF formula…
We show that the ratio of the number of near perfect matchings to the number of perfect matchings in $d$-regular strong expander (non-bipartite) graphs, with $2n$ vertices, is a polynomial in $n$, thus the Jerrum and Sinclair Markov chain…
We present a number of new results about range searching for colored (or "categorical") data: 1. For a set of $n$ colored points in three dimensions, we describe randomized data structures with $O(n\mathop{\rm polylog}n)$ space that can…
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…
We prove lower and upper bounds for the chromatic number of certain hypergraphs defined by geometric regions. This problem has close relations to conflict-free colorings. One of the most interesting type of regions to consider for this…
We give new combinatorial proofs of known almost-periodicity results for sumsets of sets with small doubling in the spirit of Croot and Sisask, whose almost-periodicity lemma has had far-reaching implications in additive combinatorics. We…
Kneser's 1955 conjecture -- proven by Lov\'asz in 1978 -- asserts that in any partition of the $k$-subsets of $\{1, 2, \dots, n\}$ into $n-2k-3$ parts, one part contains two disjoint sets. Schrijver showed that one can restrict to…
Asymptotic separation index is a parameter that measures how easily a Borel graph can be approximated by its subgraphs with finite components. In contrast to the more classical notion of hyperfiniteness, asymptotic separation index is…