Abstract: We give nontrivial bounds for the inductiveness or degeneracy of power graphs Gk of a planar graph G. This implies bounds for the chromatic number as well, since the inductiveness naturally relates to a greedy algorithm for vertex-coloring the given graph. The inductiveness moreover yields bounds for the choosability of the graph. We show that the inductiveness of a square of a planar graph G is at most $\lceil 9\Delta /5 \rceil$, for the maximum degree $\Delta$ sufficiently large, and that it is sharp. In general, we show for a fixed integer $k\geq1$ the inductiveness, the chromatic number, and the choosability of Gk to be $O(\Delta^{\lfloor k/2 \rfloor})$, which is tight.

Abstract: The strong chromatic index of graphs Mohammad Mahdian Master of Science Graduate Department of Computer Science University of Toronto A strong edge colouring of a graph G is an assignment of colours to the edges of G such that every colour class is an induced matching The minimum number of colours in such a colouring is called the strong chromatic index of G In Erd os and Ne set ril conjectured that the strong chromatic index of every graph of maximum degree is at most In this thesis we present a survey of known results related to strong edge colourings and an introduction to the probabilistic method We will use the probabilistic method to prove that the strong chromatic index of a C free graph i e a graph which does not contain a cycle as a subgraph of maximum degree is at most o ln This implies that the conjecture of Erd os and Ne set ril is true for C free graphs with large maximum degree We will show that our bound is asymptotically the best possible up to a constant multiple Also we will investigate the algorithmic aspects of the strong edge colouring problem and will prove that it is NP complete even in a very restricted setting Finally we present a list of open problems and conjectures related to strong edge colourings

Abstract: The list-chromatic index, χl′(G) of a multigraph G is the least t such that if S(A) is a set of size t for each A∈E≔E(G), then there exists a proper coloring σ of G with σ(A)∈S(A) for each A∈E. The list-chromatic index is bounded below by the ordinary chromatic index, χ′(G), which in turn is at least the fractional chromatic index, χ′*(G). In previous work we showed that the chromatic and fractional chromatic indices are asymptotically the same; here we extend this to the list-chromatic index: χl′(G)∼χ′*(G) as χl′(G)∞. The proof uses sampling from “hard-core” distributions on the set of matchings of a multigraph to go from fractional to list colorings. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 117–156, 2000

Abstract: The odd-girth of a graph is the length of a shortest odd circuit. A conjecture by Pavol Hell about circular coloring is solved in this article by showing that there is a function 𝒻(e) for each e : 0 < e < 1 such that, if the odd-girth of a planar graph G is at least 𝒻(e), then G is (2 + e)-colorable. Note that the function 𝒻(e) is independent of the graph G and e ➝ 0 if and only if 𝒻(e) ➝ ∞. A key lemma, called the folding lemma, is proved that provides a reduction method, which maintains the odd-girth of planar graphs. This lemma is expected to have applications in related problems. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 109119, 2000

Abstract: We give a new proof showing that it is NP-hard to color a 3-colorable graph using just four colors. This result is already known, but our proof is novel as it does not rely on the PCP theorem. This highlights a qualitative difference between the known hardness result for coloring 3-colorable graphs and the factor n/sup /spl epsiv// hardness for approximating the chromatic number of general graphs, as the latter result is known to imply (some form of) PCP theorem. Another aspect in which our proof is different is that using the PCP theorem we can show that 4-coloring of 3-colorable graphs remains NP-hard even on bounded-degree graphs (this hardness result does not seem to follow from the earlier reduction. We point out that such graphs can always be colored using O(1) colors by a simple greedy algorithm, while the best known algorithm for coloring (general) 3-colorable graphs requires n/sup /spl Omega/(1)/ colours. Our proof technique also shows that there is an /spl epsi//sub 0/>0 such that it is NP-hard to legally 4-color even a (1-/spl epsi//sub 0/) fraction of the edges of a 3-colorable graph.

Abstract: Let G = (V, E) be a graph and let k be a nonnegative integer. A vector c e ℤ V+ is called k-colorable iff there exists a coloring of G with k colors that assigns exactly c(v) colors to vertex v e V. Denote by χ (G) and χf(G) the chromatic number and fractional chromatic number, respectively. We prove that χ(G) = χf(G) holds for every proper circular arc graph G. For this purpose, a more general round-up property is characterized by means of a polyhedral description of all k-colorable vectors. Both round-up properties are equivalent for proper circular arc graphs. The polyhedral description is established and, as a by-product, a known coloring algorithm is generalized to multicolorings. The round-up properties do not hold for the larger classes of circular arc graphs and circle graphs, unless P = NP. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 256267, 2000

Abstract: This paper investigates applications of heuristic techniques for solving the edge coloring problem, which seeks the minimum number of colors to color the edges of a graph such that no two adjacent edges (edges that share a common vertex) get the same color. Except for special cases, such as bipartite graphs, the edge coloring problem is NP-complete. Thus, the search for exact algorithms is replaced by the investigation of approximgttion and heuristic algorithms (unless P=NP). In this work, we consider both simple graphs and multigraphs, and compare three heuristics for the edge coloring problem. The first is a greedy algorithm, and the two others are genetic algorithms: a genetic algorithm that makes use of LibGA and a grouping genetic algorithm. Our results show that the grouping genetic algorithm outperforms the greedy heuristic for many problem instances. This paper supports the clainl that the edge coloring problem is more amenable to grouping genetic algorithms.

Abstract: The Four Color Theorem says that the faces (or vertices) of a plane graph may be colored with four colors. Vizing's Theorem says that the edges of a graph with maximum degreemay be colored with� + 1 colors. In 1972, Kronk and Mitchem conjectured that the vertices, edges, and faces of a plane graph may be simultaneously colored with� + 4 colors. In this article, we give a simple proof that the conjecture is true if �≥ 6.

Abstract: The radio channel assignment problem can be cast as a graph coloring problem. Vertices correspond to transmitter locations and their labels (colors) to radio channels. The assignment of frequencies to each transmitter (vertex) must avoid interference which depends on the seperation each pair of vertices has. Two levels of interference are assumed in the problem we are concerned. Based on this channel assignment problem, we proposed a graph labelling problem which has two constraints instead of one. We consider the question of finding the minimum edge of this labelling. Several classes of graphs including one that is important to a telecommunication problem have been studied.

Abstract: We investigate a variant of on-line edge-coloring in which there is a fixed number of colors available and the aim is to color as many edges as possible. We prove upper and lower bounds on the performance of different classes of algorithms for the problem. Moreover, we determine the performance of two specific algorithms, First-Fit and Next-Fit.

Abstract: Let $G$ be a simple graph with $3\Delta (G) > |V|$. The Overfull Graph Conjecture states that the chromatic index of $G$ is equal to $\Delta (G)$, if $G$ does not contain an induced overfull subgraph $H$ with $\Delta (H) = \Delta (G)$, and otherwise it is equal to $\Delta (G) +1$. We present an algorithm that determines these subgraphs in $O(n^{5/3}m)$ time, in general, and in $O(n^3)$ time, if $G$ is regular. Moreover, it is shown that $G$ can have at most three of these subgraphs. If $2\Delta (G) \geq |V|$, then $G$ contains at most one of these subgraphs, and our former algorithm for this situation is improved to run in linear time.

Abstract: We consider graphs and multigraphs which are critical with respect to the chromatic index. In chapter 3, we give a construction of critical multigraphs with exactly 20 vertices and maximum degree k for every k>=5. This disproves the weak critical graph conjecture. In chapter 4, we give a new method, how several 4-critical multigraphs can be constructed from a given 3-critical graph. In chapter 5, we prove that the edges of every planar graph with maximum degree 7 can be colored with 7 colors. This proves one of the two open cases of Vizing's planar graph conjecture from 1965.

Abstract: We study connectivity Hamilton path and Hamil ton cycle decomposition edge and vertex col oring for geometric graphs arising from pseudoline a ne or projective and pseudocircle spherical arrangements While arrangements as geometric objects are well studied in discrete and computa tional geometry their graph theoretical properties seem to have received little attention so far In this paper we show that they provide well structured ex amples of families of planar and projective planar graphs with very interesting properties Most prominently spherical arrangements admit decom positions into two Hamilton cycles and edge color ings but other classes have interesting properties as well connectivity vertex coloring or Hamilton paths and cycles We show a number of negative results as well there are projective arrangements which cannot be vertex colored A number of con jectures and open questions accompany our results

Abstract: We give a new proof showing that it is NP-hard to color a 3-colorable graph using just 4 colors. This result is already known , [S. Khanna, N. Linial, and S. Safra, Combinatorica, 20 (2000), pp. 393--415], but our proof is novel because it does not rely on the PCP theorem, while the known one does. This highlights a qualitative difference between the known hardness result for coloring 3-colorable graphs and the factor $n^{\epsilon}$ hardness for approximating the chromatic number of general graphs, as the latter result is known to imply (some form of) PCP theorem [M. Bellare, O. Goldreich, and M. Sudan, SIAM J. Comput., 27 (1998), pp. 805--915]. Another aspect in which our proof is novel is in its use of the PCP theorem to show that 4-coloring of 3-colorable graphs remains NP-hard even on bounded-degree graphs (this hardness result does not seem to follow from the earlier reduction of Khanna, Linial, and Safra). We point out that such graphs can always be colored using O(1) colors by a simple greedy algorithm, while the best known algorithm for coloring (general) 3-colorable graphs requires $n^{\Omega(1)}$ colors. Our proof technique also shows that there is an $\varepsilon_0 > 0$ such that it is NP-hard to legally 4-color even a $(1-\varepsilon_0)$ fraction of the edges of a 3-colorable graph.

Abstract: We consider combinatorial avoidance and achievement games based on graph Ramsey theory: The players take turns in coloring edges of a graph G, each player being assigned a distinct color and choosing one so far uncolored edge per move. In avoidance games, completing a monochromatic subgraph isomorphic to another graph A leads to immediate defeat or is forbidden and the first player that cannot move loses. In the avoidance+ variant, both players are free to choose more than one edge per move. In achievement games, the first player that completes a monochromatic subgraph isomorphic to A wins. We prove that general graph Ramsey avoidance, avoidance+, and achievement endgames and several variants thereof are PSPACE-complete.

Abstract: We study the complexity of the problem 3-Colorability when restricted to those input graphs on which a given graph coloring heuristic is able to solve the problem. The heuristics we consider include the sequential algorithm traversing the vertices of the graph in various orderings (e.g., by decreasing degree or in the recursive smallest-last order) as well as Wood’s algorithm. For each heuristic considered here, we prove that the corresponding restriction of 3-Colorability remains NP-complete.

Abstract: In an ordinary edge-coloring of a graph each color appears at each vertex v at most once. A [g,f] -coloring is a generalized edge-coloring in which each color appears at each vertex v at least g(v) and at most f(v) times, where g(v) and f(v) are respectively nonnegative and positive integers assigned to v . This paper gives a linear-time algorithm to find a [g,f] -coloring of a given partial k -tree using the minimum number of colors if there exists a [g,f] -coloring.

Abstract: A coloring of graph vertices is called acyclic if the ends of each edge are colored in distinct colors and there are no two-colored cycles. Suppose each face of rank not greater thank, k ≥ 4, on a surfaceS N is replaced by the clique on the same set of vertices. Then the pseudograph obtained in this way can be colored acyclically in a set of colors whose cardinality depends linearly onN and onk. Results of this kind were known before only for 1 ≤N ≤ 2 and 3 ≤k ≤ 4.

Abstract: Fundamentals of graph coloring are introduced, and four basic alternative algorithms for coloring undirected graphs are described in J, along with programs for generating, adjacency matrices and testing them. Results are compared, and suggestions for future development are offered.