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The aim of this thesis is to investigate how the structure of a digraph affects its dichromatic number and to extend various results on undirected colouring to digraphs. In the first part of this thesis, we examine how the dichromatic…

This paper studies sufficient conditions to obtain efficient distributed algorithms coloring graphs optimally (i.e.\ with the minimum number of colors) in the LOCAL model of computation. Most of the work on distributed vertex coloring so…

A signed graph is a simple graph with two types of edges: positive and negative edges. Switching a vertex $v$ of a signed graph corresponds to changing the type of each edge incident to $v$. A homomorphism from a signed graph $G$ to another…

The defective chromatic number of a graph class $\mathcal{G}$ is the minimum integer $k$ such that for some integer $d$, every graph in $\mathcal{G}$ is $k$-colourable such that each monochromatic component has maximum degree at most $d$.…

In the past various distance based colorings on planar graphs were introduced. We turn our focus to three of them, namely $2$-distance coloring, injective coloring, and exact square coloring. A $2$-distance coloring is a proper coloring of…

In this paper we find chromatic numbers of distance graphs $G(n,3,2)$ for infinitely many n. Also we improve upper bound for $\chi(G(n,r,s))$ in large part of cases.

Given a set D of positive integers, the associated distance graph on the integers is the graph with the integers as vertices and an edge between distinct vertices if their difference lies in D. We investigate the chromatic numbers of…

Fong et al. (The game chromatic index of some trees with maximum degree four and adjacent degree-four vertices, J. Comb Optim 36 (2018) 1-12) proved that the game chromatic index of any tree $T$ of maximum degree 4 whose degree-four…

An anagram is a word of the form $WP$ where $W$ is a non-empty word and $P$ is a permutation of $W$. We study anagram-free graph colouring and give bounds on the chromatic number. Alon et al. (2002) asked whether anagram-free chromatic…

The coloring problem (i.e., computing the chromatic number of a graph) can be solved in $O^*(2^n)$ time, as shown by Bj\"orklund, Husfeldt and Koivisto in 2009. For $k=3,4$, better algorithms are known for the $k$-coloring problem.…

In a graph $G$ of maximum degree 3, let $\gamma(G)$ denote the largest fraction of edges that can be 3 edge-coloured. Rizzi \cite{Riz09} showed that $\gamma(G) \geq 1-\frac{2\strut}{\strut 3 g_{odd}(G)}$ where $g_{odd}(G)$ is the odd girth…

We show a method how to convert any graph into the binary number and vice versa. We derive upper bound for maximum number of graphs, that, have fixed number of vertices and can be colored with n colors (n is any given number). Proof for the…

We consider the problem of classifying those graphs that arise as an undirected square of an oriented graph by generalising the notion of quasi-transitive directed graphs to mixed graphs. We fully classify those graphs of maximum degree…

The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has been…

A star edge coloring of a graph is a proper edge coloring with no $2$-colored path or cycle of length four. The star chromatic index $\chi'_{st}(G)$ of $G$ is the minimum number $t$ for which $G$ has a star edge coloring with $t$ colors. We…

An edge-coloured graph $G$ is called $properly$ $connected$ if every two vertices are connected by a proper path. The $proper$ $connection$ $number$ of a connected graph $G$, denoted by $pc(G)$, is the smallest number of colours that are…

This paper examines the spectral characterizations of oriented graphs. Let $\Sigma$ be an $n$-vertex oriented graph with skew-adjacency matrix $S$. Previous research mainly focused on self-converse oriented graphs, proposing arithmetic…

We call a (not necessarily properly) edge-colored graph edge-color-avoiding connected if after the removal of edges of any single color, the graph remains connected. For vertex-colored graphs, similar definitions of color-avoiding…

We prove that up to two exceptions, every connected subcubic triangle-free graph has fractional chromatic number at most 11/4. This is tight unless further exceptional graphs are excluded, and improves the known bound on the fractional…

Given a sequence $S=(s_1,s_2,\ldots,s_p)$, $p\geq 2$, of non-decreasing integers, an $S$-packing coloring of a graph $G$ is a partition of its vertex set into $p$ disjoint sets $V_1,\ldots, V_p$ such that any two distinct vertices of $V_i$…