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Motivated by frequency assignment problems in wireless broadcast networks, Goddard, Hedetniemi, Hedetniemi, Harris, and Rall introduced the notion of $S$-packing coloring in 2008. Given a non-decreasing sequence $S = (s_1, s_2, \ldots,…
Square coloring is a variant of graph coloring where vertices within distance two must receive different colors. When considering planar graphs, the most famous conjecture (Wegner, 1977) states that $\frac32\Delta+1$ colors are sufficient…
The degree splitting problem requires coloring the edges of a graph red or blue such that each node has almost the same number of edges in each color, up to a small additive discrepancy. The directed variant of the problem requires…
Given a bipartite graph with parts $A$ and $B$ having maximum degrees at most $\Delta_A$ and $\Delta_B$, respectively, consider a list assignment such that every vertex in $A$ or $B$ is given a list of colours of size $k_A$ or $k_B$,…
In this paper we consider a variation of a recoloring problem, called the Color-Fixing. Let us have some non-proper $r$-coloring $\varphi$ of a graph $G$. We investigate the problem of finding a proper $r$-coloring of $G$, which is "the…
We develop sequential algorithms for constructing edge-colorings of graphs and multigraphs efficiently and using few colors. Our primary focus is edge-coloring arbitrary simple graphs using $d+1$ colors, where $d$ is the largest vertex…
We study the maintenance of a $(\Delta+C)$-edge-coloring ($C\ge 1$) in a fully dynamic graph $G$ with maximum degree $\Delta$. We focus on minimizing \emph{recourse} which equals the number of recolored edges per edge updates. We present a…
Graphs arising in statistical problems, signal processing, large networks, combinatorial optimization, and data analysis are often dense, which causes both computational and storage bottlenecks. One way of \textit{sparsifying} a…
This paper introduces an efficient quantum computing method for reducing special graphs in the context of the graph coloring problem. The special graphs considered include both symmetric and non-symmetric graphs where the axis passes…
In this paper, we investigate the \textsc{Grundy Coloring} problem for graphs with a cluster modulator, a structure commonly found in dense graphs. The Grundy chromatic number, representing the maximum number of colors needed for the…
Given a graph $G$, an edge-coloring is an assignment of colors to edges of $G$ such that any two edges sharing an endpoint receive different colors. By Vizing's celebrated theorem, any graph of maximum degree $\Delta$ needs at least…
The Four color problem is closely related to other branches of mathematics and practical applications. More than 20 of its reformulations are known, which connect this problem with problems of algebra, statistical mechanics and planning.…
We present a deterministic distributed algorithm that computes a $(2\Delta-1)$-edge-coloring, or even list-edge-coloring, in any $n$-node graph with maximum degree $\Delta$, in $O(\log^7 \Delta \log n)$ rounds. This answers one of the…
We present a randomized algorithm that takes as input an undirected $n$-vertex graph $G$ with maximum degree $\Delta$ and an integer $k > 3\Delta$, and returns a random proper $k$-coloring of $G$. The distribution of the coloring is…
A proper conflict-free colouring of a graph is a colouring of the vertices such that any two adjacent vertices receive different colours, and for every non-isolated vertex $v$, some colour appears exactly once on the neighbourhood of $v$.…
We consider the problem of designing deterministic graph algorithms for the model of Massively Parallel Computation (MPC) that improve with the sparsity of the input graph, as measured by the notion of arboricity. For the problems of…
In the Colored Clustering problem, one is asked to cluster edge-colored (hyper-)graphs whose colors represent interaction types. More specifically, the goal is to select as many edges as possible without choosing two edges that share an…
We revisit the problem of fault-tolerant (FT) distance preservers, when failure events in the network admit a form of correlation modeled as color faults. FT distance preservers are sparse subgraphs that preserve distances between specified…
In this paper, we study two popular variants of Graph Coloring -- Dominator Coloring and CD Coloring. In both problems, we are given a graph $G$ and a natural number $\ell$ as input and the goal is to properly color the vertices with at…
In an undirected graph, a proper (k,i)-coloring is an assignment of a set of k colors to each vertex such that any two adjacent vertices have at most i common colors. The (k,i)-coloring problem is to compute the minimum number of colors…