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The boxicity of a graph G, denoted as box(G), is the minimum dimension d for which each vertex of G can be mapped to a d-dimensional axis-parallel box in Rd such that two boxes intersect if and only if the corresponding vertices of G are adjacent. An axis-parallel box is a generalized rectangle with sides parallel to the coordinate axes. If additionally, we restrict all sides of the rectangle to be of unit length, the new parameter so obtained is called the cubicity of the graph G, denoted by cub(G). F.S. Roberts had shown that for a graph G with n vertices, box(G) ≤ and cub(G) ≤ . A minimum vertex cover of a graph G is a minimum cardinality subset S of the vertex set of G such that each edge of G has at least one endpoint in S. We show that box(G) ≤ +1 and cub(G)≤ t+ ⌈log2(n −t)⌉−1 where t is the cardinality of a minimum vertex cover. Both these bounds are tight. For a bipartite graph G, we show that box(G) ≤ and this bound is tight. We observe that there exist graphs of very high boxicity but with very low chromatic num-ber. For example, there exist bipartite (2 colorable) graphs with boxicity equal to . Interestingly, if boxicity is very close to , then the chromatic number also has to be very high. In particular, we show that if box(G) = −s, s ≥ 02, then x(G) ≥ where X(G) is the chromatic number of G. We also discuss some known techniques for findingan upper boundon the boxicityof a graph -representing the graph as the intersection of graphs with boxicity 1 (boxicity 1 graphs are known as interval graphs) and covering the complement of the graph by co-interval graphs (a co-interval graph is the complement of an interval graph).

A proper edge coloring of G =(V,E)is a map c : E → C (where C is the set of available colors ) with c(e) ≠ c(ƒ) for any adjacent edges e,f. The minimum number of colors needed to properly color the edges of G, is called the chromatic index of Gand is denoted by χ(G). A proper edge coloring c is called acyclic if there are no bichromatic cycles in the graph. In other words an edge coloring is acyclic if the union of any two color classes induces a set of paths (i.e., linear forest) in G. The acyclic edge chromatic number (also called acyclic chromatic index), denoted by a’(G), is the minimum number of colors required to acyclically edge color G. The primary motivation for this thesis is the following conjecture by Alon, Sudakov and Zaks [7] (and independently by Fiamcik [22]): Acyclic Edge Coloring Conjecture: For any graph G, a’ (G) ≤ Δ(G)+2. The following are the main results of the thesis: 1 From a result of Burnstein [16], it follows that any subcubic graph can be acyclically edge colored using at most 5 colors. Skulrattankulchai [38] gave a polynomial time algorithm to color a subcubic graph using Δ + 2 = 5 colors. We proved that any non-regular subcubic graph can be acyclically colored using only 4 colors. This result is tight. This also implies that the fifth color, when needed is required only for one edge. 2 Let G be a connected graph on n vertices, m ≤ 2n - 1 edges and maximum degree Δ ≤ 4, then a’ (G) ≤ 6. This implies that graph of maximum degree 4 are acyclically edge colorable using at most 7 colors. 3 The earliest result on acyclic edge coloring of 2-degenerate graphs was by Caro and Roditty [17], where they proved that a’ (G) ≤ Δ + k - 1, where k is the maximum edge connectivity, defined as k = maxu,vε V(G)λ(u,v), where λ(u,v)is the edge-connectivity of the pair u,v. Note that here k can be as high as Δ. Muthu,Narayanan and Subramanian [34] proved that a’ (G) ≤ Δ + 1for outerplanar graphs which are a subclass of 2-degenerate graphs and posed the problem of proving the conjecture for 2-degenerate graphs as an open problem. In fact they have also derived an upper bound of Δ+1 for series-parallel graphs [35], which is a slightly bigger subclass of 2-degenerate graphs. We proved that 2-degenerate graphs are Δ+1colorable. 1 Fiedorowicz, Hauszczak and Narayanan [24] gave an upper bound of 2Δ+29 for planar graphs. Independently Hou, Wu, GuiZhen Liu and Bin Liu [29] gave an upper bound of max(2Δ - 2,Δ+ 22). We improve this upper bound to Δ+12, which is the best known bound at present. 2 Fiedorowicz, Hauszczak and Narayanan [24] gave an upper bound of Δ+6for triangle free planar graphs. We improve the bound to Δ+3, which is the best known bound at present. 3 We have also worked on lower bounds. Alon et.al. [7], along with the acyclic edge coloring conjecture, also made an auxiliary conjecture stating that Complete graphs of 2n vertices are the only class of regular graphs which require Δ+2colors. We disproved this conjecture by showing infinite classes of regular graphs other than Complete Graphs which require Δ+2colors. Apart from the above mentioned results, this thesis also contributes to the acyclic edge coloring literature by introducing new techniques like Recoloring, Color Exchange (exchanging colors of adjacent edges), circular shifting of colors on adjacent edges (derangement of colors). These techniques turn out to be very useful in proving upper bounds on the acyclic edge chromatic number.

This thesis touches three different topics in graph theory, namely, rainbow colouring, product dimension and boxicity. Rainbow colouring An edge colouring of a graph is called a rainbow colouring, if every pair of vertices is connected by atleast one path in which no two edges are coloured the same. The rainbow connection number of a graph is the minimum number of colours required to rainbow colour it. In this thesis we give upper bounds on rainbow connection number based on graph invariants like minimum degree, vertex connectivity, and radius. We also give some computational complexity results for special graph classes. Product dimension The product dimension or Prague dimension of a graph G is the smallest natural number k such that G is an induced subgraph of a direct product of k complete graphs. In this thesis, we give upper bounds on the product dimension for forests, bounded tree width graphs and graphs of bounded degeneracy. Boxicity and cubicity The boxicity (cubicity of a graph G is the smallest natural number k such that G can be represented as an intersection graph of axis-parallel rectangular boxes(axis-parallel unit cubes) in Rk .In this thesis, we study the boxicity and the cubicity of Cartesian, strong and direct products of graphs and give estimates on the boxicity and the cubicity of a product graph based on invariants of the component graphs. Separation dimension The separation dimension of a hypergraph H is the smallest natural number k for which the vertices of H can be embedded in Rk such that any two disjoint edges of H can be separated by a hyper plane normal to one of the axes. While studying the boxicity of line graphs, we noticed that a box representation of the line graph of a hypergraph has a nice geometric interpretation. Hence we introduced this new parameter and did an extensive study of the same.

In this thesis, we focus on the study of computational and combinatorial problems on various geometric proximity graphs. Delaunay and Gabriel graphs are widely studied geometric proximity structures. These graphs have been extensively studied for their applications in wireless networks. Motivated by the applications in localized wireless routing, relaxed versions of these graphs known as Locally Delaunay Graphs (LDGs) and Locally Gabriel Graphs(LGGs) were proposed. A geometric graph G=(V,E)is called a Locally Gabriel Graph if for every( u,v) ϵ E the disk with uv as diameter does not contain any neighbor of u or v in G. Thus, two edges (u, v) and(u, w)where u,v,w ϵ V conflict with each other if ∠uwv ≥ or ∠uvw≥π and cannot co-exist in an LGG. We propose another generalization of LGGs called Generalized locally Gabriel Graphs(GLGGs)in the context when certain edges are forbidden in the graph. For a given geometric graph G=(V,E), we define G′=(V,E′) as GLGG if G′is an LGG and E′⊆E. Unlike a Gabriel Graph ,there is no unique LGG or GLGG for a given point set because no edge is necessarily included or excluded. This property allows us to choose an LGG/GLGG that optimizes a parameter of interest in the graph. While Gabriel graphs are planar graphs, there exist LGGs with super linear number of edges. Also, there exist point sets where a Gabriel graph has dilation of Ω(√n)and there exist LGGs on the same point sets with dilation O(1). We study these graphs for various parameters like edge complexity(the maximum number of edges in these graphs),size of an independent set and dilation. We show that computing an edge maximum GLGG for a given problem instance is NP-hard and also APX-hard. We also show that computing an LGG on a given point set with minimum dilation is NP-hard. Then, we give an algorithm to verify whether a given geometric graph G=(V,E)is an LGG with running time O(ElogV+ V). We show that any LGG on n vertices has an independent set of size Ω(√nlogn). We show that there exists point sets with n points such that any LGG on it has dilation Ω(√n) that matches with the known upper bound. Then, we study some greedy heuristics to compute LGGs with experimental evaluation. Experimental evaluations for the points on a uniform grid and random point sets suggest that there exist LGGs with super-linear number of edges along with an independent set of near-linear size. Unit distance graphs(UDGs) are well studied geometric graphs. In this graph, an edge exists between two points if and only if the Euclidean distance between the points is unity. UDGs have been studied extensively for various properties most notably for their edge complexity and chromatic number. These graphs have also been studied for various special point sets most notably the case when the points are in convex position. Note that the UDGs form a sub class of the LGGs. UDGs/LGGs on convex point sets have O(nlogn) edges. The best known lower bound on the edge complexity of these graphs is 2n−7 when all the points are in convex position. A bipartite graph is called an ordered bipartite graph when the vertex set in each partition has a total order on its vertices. We introduce a family of ordered bipartite graphs with restrictions on some paths called path restricted ordered bi partite graphs (PRBGs)and show that their study is motivated by LGGs and UDGs on convex point sets. We show that a PRBG can be extracted from the UDGs/LGGs on convex point sets. First, we characterize a special kind of paths in PRBGs called forward paths, then we study some structural properties of these graphs. We show that a PRBG on n vertices has O(nlogn) edges and the bound is tight. It gives an alternate proof of O(nlogn)upper bound for the maximum number of edges in UDGs/LGGs on convex point sets. We study PRBGs with restrictions to the length of the forward paths and show an improved bound on the edge complexity when the length of the longest forward path is bounded. Then, we study the hierarchical structure amongst these graphs classes. Notably, we show that the class of UDGs on convex point sets is a strict sub class of LGGs on convex point sets.

The lower and upper independent, domination and irredundant numbers of the graph G = (V, E) are denoted by i ( G) , f3 ( G), 'Y ( G), r ( G), ir ( G) and IR ( G) respectively. These six numbers are called the domination parameters. For each of these parameters n:, we define six types of criticality. The graph G is n:-critical (n:+ -critical) if the removal of any vertex of G causes n: (G) to decrease (increase), G is n:-edge-critical (n:+-edge-critical) if the addition of any missing edge causes n: (G) to decrease (increase), and G is Ir-ER-critical (n:- -ER-critical) if the removal of any edge causes n: (G) to increase (decrease). For all the above-mentioned parameters n: there exist graphs which are n:-critical, n:-edge-critical and n:-ER-critical. However, there do not exist any n:+-critical graphs for n: E {ir,"f,i,/3,IR}, no n:+-edge-critical graphs for n: E {ir,"f,i,/3} and non:--ER-critical graphs for: E {'Y,/3,r,IR}. Graphs which are "I-critical, i-critical, "I-edge-critical and i-edge-critical are well studied in the literature. In this thesis we explore the remaining types of criticality. We commence with the determination of the domination parameters of some wellknown classes of graphs. Each class of graphs we consider will turn out to contain a subclass consisting of graphs that are critical according to one or more of the definitions above. We present characterisations of "I-critical, i-critical, "I-edge-critical and i-edge-critical graphs, as well as ofn:-ER-critical graphs for n: E {/3,r,IR}. These characterisations are useful in deciding which graphs in a specific class are critical. Our main results concern n:-critical and n:-edge-critical graphs for n: E {/3, r, IR}. We show that the only /3-critical graphs are the edgeless graphs and that a graph is IRcritical if and only if it is r-critical, and proceed to investigate the r-critical graphs which are not /3-critical. We characterise /3-edge-critical and r-edge-critical graphs and show that the classes of IR-edge-critical and r-edge-critical graphs coincide. We also exhibit classes of r+ -critical, r+ -edge-critical and i- -ER-critical graphs. / Mathematical Sciences / D. Phil. (Mathematics)

The lower and upper independent, domination and irredundant numbers of the graph G = (V, E) are denoted by i ( G) , f3 ( G), 'Y ( G), r ( G), ir ( G) and IR ( G) respectively. These six numbers are called the domination parameters. For each of these parameters n:, we define six types of criticality. The graph G is n:-critical (n:+ -critical) if the removal of any vertex of G causes n: (G) to decrease (increase), G is n:-edge-critical (n:+-edge-critical) if the addition of any missing edge causes n: (G) to decrease (increase), and G is Ir-ER-critical (n:- -ER-critical) if the removal of any edge causes n: (G) to increase (decrease). For all the above-mentioned parameters n: there exist graphs which are n:-critical, n:-edge-critical and n:-ER-critical. However, there do not exist any n:+-critical graphs for n: E {ir,"f,i,/3,IR}, no n:+-edge-critical graphs for n: E {ir,"f,i,/3} and non:--ER-critical graphs for: E {'Y,/3,r,IR}. Graphs which are "I-critical, i-critical, "I-edge-critical and i-edge-critical are well studied in the literature. In this thesis we explore the remaining types of criticality. We commence with the determination of the domination parameters of some wellknown classes of graphs. Each class of graphs we consider will turn out to contain a subclass consisting of graphs that are critical according to one or more of the definitions above. We present characterisations of "I-critical, i-critical, "I-edge-critical and i-edge-critical graphs, as well as ofn:-ER-critical graphs for n: E {/3,r,IR}. These characterisations are useful in deciding which graphs in a specific class are critical. Our main results concern n:-critical and n:-edge-critical graphs for n: E {/3, r, IR}. We show that the only /3-critical graphs are the edgeless graphs and that a graph is IRcritical if and only if it is r-critical, and proceed to investigate the r-critical graphs which are not /3-critical. We characterise /3-edge-critical and r-edge-critical graphs and show that the classes of IR-edge-critical and r-edge-critical graphs coincide. We also exhibit classes of r+ -critical, r+ -edge-critical and i- -ER-critical graphs. / Mathematical Sciences / D. Phil. (Mathematics)

Submitted by (lucia.rodrigues@ufrpe.br) on 2017-03-28T12:41:18Z No. of bitstreams: 1 Adriana Priscila de Brito.pdf: 1439366 bytes, checksum: 6c39b441ca6cf64e146c11f1a5822457 (MD5) / Made available in DSpace on 2017-03-28T12:41:18Z (GMT). No. of bitstreams: 1 Adriana Priscila de Brito.pdf: 1439366 bytes, checksum: 6c39b441ca6cf64e146c11f1a5822457 (MD5) Previous issue date: 2014-08-08 / This presentation provides an introduction to graph theory, making the connection between some of its concepts and the and characterization of Regular Polyhedra. Special emphasis will be given to the study of Eulerian graphs, Euler's Formula, Graphs and Planar Graphs Platonic. Finally, a proposed instructional sequence that focuses on introducing the concept of the graph elementary school students, making connections with the regular polyhedra is presented. / O presente trabalho tem como objetivo principal apresentar uma introdução à Teoria dos Grafos, fazendo a ligação entre alguns dos seus conceitos e a caracterização dos Poliedros Regulares. Será dada uma ênfase especial ao estudo dos Grafos Eulerianos, da Fórmula de Euler, dos Grafos Planares e dos Grafos Platônicos. Por fim, será apresentada uma proposta de sequência didática que tem como foco introduzir o conceito de grafo a alunos do ensino básico, fazendo ligações com os Poliedros Regulares.

In this thesis we derive some basic properties of graphs G embedded in a surface determining a link diagram D(G), having a specified number μ(D(G)) of components. ( The relationship between the graph and the link diagram comes from the tangle which replaces each edge of the graph). Firstly, we prove that μ (D(G)) ≤ f (G) + 2g, where f (G) is the number of faces in the embedding of G and g is the genus of the surface. Then we focus on the extremal case, where μ (D(G)) = f (G) + 2g. We note that μ (D(G)) does not change when undergoing graph Reidemeister moves or embedded ∆ ↔ Y exchanges. It is also useful that μ(D(G)) changes only very slightly when an edge is added to the graph. We finish with some observations on other possible values of μ(D(G)). We comment on two cases: when μ = 1, and the Petersen and Heawood families of graphs. These two families are obtained from K6 and K7 respectively by using ∆ ↔ Y exchanges.