Small cycle cover, group coloring with related problems

Bondy conjectured that if G is a simple 2-connected graph with n ≥ 3 vertices, then the edges of G can be covered by at most 2n-33 cycles. In Chapter 2, a result on small cycle cover is obtained and we also show that the result is as best as possible.;Thomassen conjectured that every 4-connected line graph is hamiltonian. In Chapters 3 and 4, we apply Catlin\u27s reduction method to study cycles in line graphs. Results about hamiltonian connectivity of line graphs and 3-edge-connected graphs are obtained. Several former results are extended.;Jaeger, Linial, Payan and Tarsi introduced group coloring in 1992 and proved that the group chromatic number for every planar graph is at most 6. It is shown that the bound 6 can be decreased to 5. Jaeger, Linial, Payan and Tarsi also proved that the group chromatic number for every planar graph with girth at least 4 is at most 4. Chapters 5 and 6 are devoted to the study of group coloring of graphs.;The concept of list coloring, choosability and choice number was introduced by Erdos, Rubin and Taylor in 1979 and Vizing in 1976. Alon and Tarsi proved that every bipartite planar graph is 3-choosable. Thomassen showed that every planar graph is 5-choosable and that every planar graph with girth at least 5 is 3-choosable. In Chapter 7, results on list coloring are obtained, extending a former result of Thomassen

Edge-choosability of Planar Graphs

According to the List Colouring Conjecture, if G is a multigraph then χ' (G)=χl' (G) . In this thesis, we discuss a relaxed version of this conjecture that every simple graph G is edge-(∆ + 1)-choosable as by Vizing’s Theorem ∆(G) ≤χ' (G)≤∆(G) + 1. We prove that if G is a planar graph without 7-cycles with ∆(G)≠5,6 , or without adjacent 4-cycles with ∆(G)≠5, or with no 3-cycles adjacent to 5-cycles, then G is edge-(∆ + 1)-choosable

Planar graphs without 3-cycles and with 4-cycles far apart are 3-choosable

A graph G is said to be L-colourable if for a given list assignment L = {L(v)|v ∈ V (G)} there is a proper colouring c of G such that c(v) ∈ L(v) for all v in V (G). If G is L-colourable for all L with |L(v)| ≥ k for all v in V (G), then G is said to be k-choosable. This paper focuses on two different ways to prove list colouring results on planar graphs. The first method will be discharging, which will be used to fuse multiple results into one theorem. The second method will be restricting the lists of vertices on the boundary and applying induction, which will show that planar graphs without 3- cycles and 4-cycles distance 8 apart are 3-choosable

List-coloring and sum-list-coloring problems on graphs

Graph coloring is a well-known and well-studied area of graph theory that has many applications. In this dissertation, we look at two generalizations of graph coloring known as list-coloring and sum-list-coloring. In both of these types of colorings, one seeks to first assign palettes of colors to vertices and then choose a color from the corresponding palette for each vertex so that a proper coloring is obtained. A celebrated result of Thomassen states that every planar graph can be properly colored from any arbitrarily assigned palettes of five colors. This result is known as 5-list-colorability of planar graphs. Albertson asked whether Thomassen\u27s theorem can be extended by precoloring some vertices which are at a large enough distance apart. Hutchinson asked whether Thomassen\u27s theorem can be extended by allowing certain vertices to have palettes of size less than five assigned to them. In this dissertation, we explore both of these questions and answer them in the affirmative for various classes of graphs. We also provide a catalog of small configurations with palettes of different prescribed sizes and determine whether or not they can always be colored from palettes of such sizes. These small configurations can be useful in reducing certain planar graphs to obtain more information about their structure. Additionally, we look at the newer notion of sum-list-coloring where the sum choice number is the parameter of interest. In sum-list-coloring, we seek to minimize the sum of varying sizes of palettes of colors assigned the vertices of a graph. We compute the sum choice number for all graphs on at most five vertices, present some general results about sum-list-coloring, and determine the sum choice number for certain graphs made up of cycles