Related papers: Coloring bipartite graphs with semi-small list siz…
List packing is a notion that was introduced in 2021 (by Cambie et al.). The list packing number of a graph $G$, denoted $\chi_{\ell}^*(G)$, is the least $k$ such that for any list assignment $L$ that assigns $k$ colors to each vertex of…
In this work, we continue the study of vertex colorings of graphs, in which adjacent vertices are allowed to be of the same color as long as each monochromatic connected component is of relatively small cardinality. We focus on colorings…
A matching in a graph is uniquely restricted if no other matching covers exactly the same set of vertices. This notion was defined by Golumbic, Hirst, and Lewenstein and studied in a number of articles. Our contribution is twofold. We…
Karonski, Luczak, and Thomason (2004) conjectured that, for any connected graph G on at least three vertices, there exists an edge weighting from {1,2,3} such that adjacent vertices receive different sums of incident edge weights.…
This paper proves limit theorems for the number of monochromatic edges in uniform random colorings of general random graphs. These can be seen as generalizations of the birthday problem (what is the chance that there are two friends with…
The online list coloring is a widely studied topic in graph theory. A graph $G$ is 2-paintable if we always have a strategy to complete a coloring in an online list coloring of $G$ in which each vertex has a color list of size 2. In this…
$(1^a, 2^b)$-coloring is the problem of partitioning the vertex set of a graph into $a$ independent sets and $b$ 2-independent sets. This problem was recently introduced by Choi and Liu. We study the computational complexity and extremal…
List coloring generalizes graph coloring by requiring the color of a vertex to be selected from a list of colors specific to that vertex. One refinement of list coloring, called choosability with separation, requires that the intersection…
An equitable coloring of a graph $G$ is a proper vertex coloring of $G$ such that the sizes of any two color classes differ by at most one. In the paper, we pose a conjecture that offers a gap-one bound for the smallest number of colors…
We prove that there exist perfect graphs of arbitrarily large clique-chromatic number. These graphs can be obtained from cobipartite graphs by repeatedly gluing along cliques. This negatively answers a question raised by Duffus, Sands,…
We describe several graphs with arbitrarily large rankwidth (or equivalently with arbitrarily large cliquewidth). Korpelainen, Lozin, and Mayhill [Split permutation graphs, Graphs and Combinatorics, 30(3):633-646, 2014] proved that there…
Gy\'arf\'as famously showed that in every $r$-coloring of the edges of the complete graph $K_n$, there is a monochromatic connected component with at least $\frac{n}{r-1}$ vertices. A recent line of study by Conlon, Tyomkyn, and the second…
We consider the problem of existence of perfect $2$-colorings (equitable $2$-partitions) of Hamming graphs with given parameters. We start with conditions on parameters of graphs and colorings that are necessary for their existence. Next we…
We consider $m$-colorings of the edges of a complete graph, where each color class is defined semi-algebraically with bounded complexity. The case $m = 2$ was first studied by Alon et al., who applied this framework to obtain surprisingly…
Let $r \geqslant 0$ and $k \geqslant 1$ be integers. We say that a graph $G$ has an $r$-equitable $k$-coloring if there exists a proper $k$-coloring of $G$ such that the sizes of any two color classes differ by at most $r$. The least $k$…
A strong edge coloring of a graph $G$ is a proper edge coloring in which each color class is an induced matching of $G$. In 1993, Brualdi and Quinn Massey proposed a conjecture that every bipartite graph without $4$-cycles and with the…
We propose a random bipartite graph with weights assigned to both parts of the vertex sets. Edges are formed independently with probabilities that depend on these weights. This bipartite graph naturally gives rise to a random intersection…
Ramsey's Theorem guarantees for every graph H that any 2-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of H. In 1962, Erdos conjectured that the random 2-edge-coloring minimizes the number of…
In "Bipartite minors" [Journal of Combinatorial Theory, Series B, 2016], Chudnovsky et al. introduced the bipartite minor relation, a quasi-order on the class of bipartite graphs somewhat analogous the minor relation on general graphs and…
Alon and Krivelevich (SIAM J. Discrete Math. 15(2): 211-227 (2002)) show that if a graph is {\epsilon}-far from bipartite, then the subgraph induced by a random subset of O(1/{\epsilon}) vertices is bipartite with high probability. We…