A graph function $f$ is a mapping which assigns each graph $H$ a positive integer $f(H) leq |V(H)|$ such that $f(H)=f(H')$ if $H$ and $H'$ are isomorphic. Given a graph function $f$ and a graph $G$, an $f$-colouring of $G$ is a mapping $c: V(G) o mathbb{N}$ such that every subgraph $H$ of $G$ receives at least $f(H)$ colours, i.e., $|c(H)| geq f(H)$. The $f$-chromatic number, $chi(f,G)$, is the minimum number of colours used in an $f$-colouring of $G$. The $f$-chromatic number of a graph is a natural generalization of the chromatic number of a graph introduced by Nev{s}etv{r}il and Ossena de Mendez. Intuitively, we would like to colour the vertices of a graph $G$ with minimum number of colours subject to the constraint that the number of colours assigned to certain subgraphs must be large enough. The original chromatic number of a graph and many of its generalizations are of this nature. For example, the chromatic number of a graph is the least number of colours needed to colour the vertices of the graph so that any subgraph isomorphic to $K_2$ receives $2$ colours. Acyclic chromatic number of a graph is the least number of colours needed to colour the vertices of the graph so that any subgraph isomorphic to $K_2$ receives $2$ colours, and each cycle receives at least $3$ colours. This thesis studies the game version of $f$-colouring of graphs. Suppose $G$ is a graph and $X$ is a set of colours. Two players, Alice and Bob, take turns colour the vertices of $G$ with colours from the set $X$. A partial colouring of $G$ is legal (with respect to graph function $f$) if for any subgraph $H$ of $G$, the sum of the number of colours used in $H$ and the number of uncoloured vertices of $H$ is at least $f(H)$. Both Alice and Bob must colour legally (i.e., the partial colouring produced needs to be legal). The game ends if either all the vertices are coloured or there are uncoloured vertices but there is no legal colour for any of the uncoloured vertices. In the former case, Alice wins the game. In the latter case, Bob wins the game. The $f$-game chromatic number of $G$, $chi_g(f, G)$, is the least number of colours that the colour set $X$ needs to contain so that Alice has a winning strategy. Observe that if $|X| = |V(G)|$, then Alice always wins. So the parameter $chi_g(f,G)$ is well-defined. We define the $f$-game chromatic index on a graph $G$, $chi'(f,G)$, to be the $f$-game chromatic number on the line graph of $G$. A natural problem concerning the $f$-game chromatic number of graphs is for which graph functions $f$, the $f$-game chromatic number of $G$ is bounded by a constant for graphs $G$ from some natural classes of graphs. In case the $f$-game chromatic number of a class ${cal K}$ of graphs is bounded by a constant, one would like to find the smallest such constant. This thesis studies the $f$-game chromatic number or index for some special classes of graphs and for some special graph functions $f$. The graph functions $f$ considered are the following graph functions: 1. The $d$-relaxed function, ${ m Rel}_d$, is defined as ${ m Rel}_d(K_{1,d+1})=2$ and ${ m Rel}_d(H)=1$ otherwise. 2. The acyclic function, ${ m Acy}$, is defined as ${ m Acy}(K_2)=2$ and ${ m Acy}(C_n)=3$ for any $n geq 3$ and ${ m Acy}(H)=1$ otherwise. 3. The $i$-path function, ${ m Path}_i$, is defined as ${ m Path}_i(K_2)=2$ and ${ m Path}_i(P_i)=3$ and ${ m Path}_i(H)=1$ otherwise, where $P_i$ is the path on $i$ vertices. The classes of graphs considered in the thesis are outerplanar graphs, forests and the line graphs of $d$-degenerate graphs. In Chapter 2, we discuss the acyclic game chromatic number of outerplanar graphs. It is proved that for any outerplanar graph $G$, $chi_g({ m Acy},G) leq 7$. On the other hand, there is an outerplanar graph $G$ for which $chi_g({ m Acy},G) = 6$. So the best upper bound for $chi_g({ m Acy},G)$ for outerplanar graphs is either $6$ or $7$. Moreover, we show that for any integer $n$, there is a series-parallel graph $G_n$ with $chi_g({ m Acy}, G_n) > n$. In Chapter 3, we discuss the ${ m Path}_i$-game chromatic number for forests. It is proved that if $i geq 8$, then for any forest $F$, $chi_g({ m Path}_i, F) leq 9$. On the other hand, if $i leq 6$, then for any integer $k$, there is a tree $T$ such that $chi_g({ m Path}_i, T) geq k$. Chapter 4 studies the $d$-relaxed game chromatic indexes of $k$-degenerated graphs. It is proved that if $d geq 2k^2 + 5k-1$ and $G$ is $k$-degenerated, then $chi'_{ m g}({ m Rel}_d,G) leq 2k + frac{(Delta(G)+k-1)(k+1)}{d-2k^2-4k+2}$. On the other hand, for any positive integer $ d leq Delta-2$, there is a tree $T$ with maximum degree $Delta$ for which $chi'_g({ m Rel}_d, T) geq frac{2Delta}{d+3}$. Moreover, we show that $chi'_g({ m Rel}_d, G) leq 2$ if $d geq 2k + 2lfloor frac{Delta(G)-k}{2} floor +1$ and $G$ is a $k$-degenerated graph.

Let S be a set of k distinct elements. An oriented k coloring of an oriented graph D is a mapping f:V(D)¡÷S such that (i) if xy is conatined in A(D), then f(x)¡Úf(y) and (ii) if xy,zt are conatined in A(D) and f(x)=f(t), then f(y)¡Úf(z). The oriented chromatic number Xo(D) of an oriented graph D is defined as the minimum k where there exists an oriented k-coloring of D. For an undirected graph G, let O(G) be the set of all orientations D of G. We define the oriented chromatic number Xo(G) of G to be the maximum of Xo(D) over D conatined by O(G). In this thesis, we determine the oriented chromatic number of complete bipartite graphs and complete k-partite graphs. A grid G(m,n) is a graph with the vertex set V(G(m,n))={(i,j) | 1¡Øi¡Øm,1¡Øj¡Øn} and the edge set E(G(m,n))={(i,j)(x,y) | (i=x+1 and j=y) or (i=x and j=y+1)}. Fertin, Raspaud and Roychowdhury [3] proved Xo(G(4,5))¡Ù7 by computer programs. Here, we give a proof of Xo(D(5,6)=7 where D(5,6) is the orientation of G(5,6).

This thesis discusses graph marking game and graph colouring game. Suppose G=(V, E) is a graph. A marking game on G is played by two players, Alice and Bob, with Alice playing first. At the start of the game all vertices are unmarked. A play by either player consists of marking an unmarked vertex. The game ends when all vertices are marked. For each vertex v of G, write t(v)=j if v is marked at the jth step. Let s(v) denote the number of neighbours u of v for which t(u) < t(v), i.e., u is marked before v. The score of the game is $$s = 1+ max_{v in V} s(v).$$ Alice's goal is to minimize the score, while Bob's goal is to maximize it. The game colouring number colg(G) of G is the least s such that Alice has a strategy that results in a score at most s. Suppose r geq 1, d geq 0 are integers. In an (r, d)-relaxed colouring game of G, two players, Alice and Bob, take turns colouring the vertices of G with colours from a set X of r colours, with Alice having the first move. A colour i is legal for an uncoloured vertex x (at a certain step) if after colouring x with colour i, the subgraph induced by vertices of colour i has maximum degree at most d. Each player can only colour an uncoloured vertex with a legal colour. Alice's goal is to have all the vertices coloured, and Bob's goal is the opposite: to have an uncoloured vertex without legal colour. The d-relaxed game chromatic number of a graph G, denoted by $chi_g^{(d)}(G)$ is the least number r so that when playing the (r, d)-relaxed colouring game on G, Alice has a winning strategy. If d=0, then the parameter is called the game chromatic number of G and is also denoted by $chi_g(G)$. This thesis obtains upper and lower bounds for the game colouring number and relaxed game chromatic number of various classes of graphs. Let colg(PT_k) and colg(P) denote the maximum game colouring number of partial k trees and the maximum game colouring number of planar graphs, respectively. In this thesis, we prove that colg(PT_k) = 3k+2 and colg(P) geq 11. We also prove that the game colouring number colg(G) of a graph is a monotone parameter, i.e., if H is a subgraph of G, then colg(H) leq colg(G). For relaxed game chromatic number of graphs, this thesis proves that if G is an outerplanar graph, then $chi_g^{(d)}(G) leq 7-t$ for $t= 2, 3, 4$, for $d geq t$, and $chi_g^{(d)}(G) leq 2$ for $d geq 6$. In particular, the maximum $4$-relaxed game chromatic number of outerplanar graphs is equal to $3$. If $G$ is a tree then $chi_ g^{(d)}(G) leq 2$ for $d geq 2$.

In this thesis we focus on the investigation of the relation between the the chip-firing and fractional coloring. Since chi_{f}(G)=inf {n/k : G is homomorphic to K(n,k)}, we find that G has an (n,k)-periodic sequence for some configuration if and only if G is homomorphic to K(n,k). Then we study the periodic configurations for the Kneser graphs. Finally, we try to evaluate the number of chips of the periodic configurations for K(n,k).

This thesis studies the circular chromatic number of Kneser graphs. It was known that if m is greater than 2n^{2}(n-1), then the Kneser graph KG(m,n) has its circular chromatic number equal its chromatic number . In particular, if n = 3, then KG(m,3) has its circular chromatic number equal its chromatic number when m is greater than 36. In this thesis, we improve this result by proving that if m is greaer than 24, then chi_c(KG(m,3)) = chi(KG(m,3)).

Abstract For a set D of positive integers, the integral distance graph G(Z, D) is the graph with vertex set Z and edge set { xy : x, y 2 Z and | x − y | 2 D } . An integral distance graph G(Z, D) is called ¡§locally dense¡¨ if the clique size of G(Z, D) is not less to | D | . This paper acterizes locally dense integral distance graphs and determine their chromatic numbers.

This thesis discusses the game chromatic number of Halin graphs. We shall prove that with a few exceptions, all Halin graphs have game chromatic number 4.

Given an n-vertex graph Gand two positive integers d, k is an element of N, the (d, kn)-differential coloring problem asks for a coloring of the vertices of G(if one exists) with distinct numbers from 1 to kn(treated as colors), such that the minimum difference between the two colors of any adjacent vertices is at least d. While it was known that the problem of determining whether a general graph is (2, n)-differential colorable is NP-complete, our main contribution is a complete characterization of bipartite, planar and outerplanar graphs that admit (2, n)-differential colorings. For practical reasons, we also consider color ranges larger than n, i.e., k > 1. We show that it is NP-complete to determine whether a graph admits a (3, 2n)-differential coloring. The same negative result holds for the (left perpendicular 2n/3 right pendicular, 2n)-differential coloring problem, even in the case where the input graph is planar.

This thesis explores several of the major results in list-coloring in an expository fashion. As a specialization of list coloring, the sum list chromatic number is explored in detail. Ultimately, the thesis is designed to motivate the discussion of coloring problems and, hopefully, interest the reader in the branch of coloring problems in graph theory.

This thesis discusses colouring, circular list colouring and adapted game colouring of graphs. For colouring, this thesis obtains a sufficient condition for a planar graph to be 3-colourable. Suppose G is a planar graph. Let H_G be the graph with vertex set V (H_G) = {C : C is a cycle of G with |C| ∈ {4, 6, 7}} and edge set E(H_G) = {CiCj : Ci and Cj have edges in common}. We prove that if any 3-cycles and 5-cycles are not adjacent to i-cycles for 3 ≤ i ≤ 7, and H_G is a forest, then G is 3-colourable. For circular consecutive choosability, this thesis obtains a basic relation among chcc(G), X(G) and Xc(G) for any finite graph G. We show that for any finite graph G, X(G) − 1 ≤ chcc(G) < 2 Xc(G). We also determine the value of chcc(G) for complete graphs, trees, cycles, balanced complete bipartite graphs and some complete multi-partite graphs. Upper and lower bounds for chcc(G) are given for some other classes of graphs. For adapted game chromatic number, this thesis studies the adapted game chromatic number of various classes of graphs. We prove that the maximum adapted game chromatic number of trees is 3; the maximum adapted game chromatic number of outerplanar graphs is 5; the maximum adapted game chromatic number of partial k-trees is between k + 2 and 2k + 1; and the maximum adapted game chromatic number of planar graphs is between 6 and 11. We also give upper bounds for the Cartesian product of special classes of graphs, such as the Cartesian product of partial k-trees and outerplanar graphs, or planar graphs.