چكيده فارسي :
يك رنگ‌آميزي رأسي از يك گراف، غيرتكراري گفته مي‌شود، هرگاه براي هر مسير مرتبه زوج از گراف، دنباله‌ي رنگ‌هاي رأس‌ها روي نيمه اول مسير متفاوت از دنباله‌ي رنگ‌هاي رأس‌ها روي نيمه دوم مسير باشد؛ به بيان ديگر، هيچ مسيري در گراف با الگوي رنگ به شكل xx (كه x دنباله‌اي از رنگ‌هاي رأس‌ها است)، وجود نداشته باشد. كم‌ترين تعداد رنگ‌هاي مورد نياز براي رنگ‌آميزي رأسي غيرتكراري گراف، عدد رنگي غيرتكراري گراف ناميده مي‌شود. مهم‌ترين مسأله در زمينه‌ي رنگ‌آميزي غيرتكراري، يافتن كران بالا براي عدد رنگي غيرتكراري كلاس‌هاي مختلف گراف‌ها است (به عنوان مثال، گراف‌هاي با ماكزيمم درجه كران‌دار، گراف‌هاي با عرض درختي كران‌دار و گراف‌هاي مسطح). اين مدل رنگ‌آميزي همچنين در انواع رنگ‌آميزي يالي، رنگ‌آميزي ليستي، رنگ‌آميزي كلي و رنگ‌آميزي غيرتكراري وجهي مطالعه شده است. در اين پايان‌نامه، رنگ‌آميزي رأسي غيرتكراري گراف‌ها مطالعه و نتايج موجود در مورد آن بيان شده است. به بيان دقيق‌تر، نتايج موجود در مورد ساختار گراف‌هايي كه عدد رنگي غيرتكراري كران‌دار دارند و همچنين كران‌هاي يافته شده روي عدد رنگي غيرتكراري كلاس‌هاي مختلف گراف‌ها ارائه شده است.
چكيده انگليسي :
```A vertex coloring of a graph is nonrepetitive if for every path of even order, the sequence of colors on the first half of the path is different from the sequence of colors on the second half; in other words, a vertex coloring of a graph is nonrepetitive if the graph contains no path that has a color pattern of the form xx (where x is a sequence of colors). In 1906, Thue constructed arbitrarily long sequences on three different symbols with no repeated consecutive blocks (a block of a sequence is a subsequence consisting of consecutive terms of the sequence). Such a sequence for which no two adjacent blocks are exactly the same is called a nonrepetitive sequence (also is called a Thue sequence or a square-free sequence). For instance, the words minimize and total are nonrepetitive, while the words repetitive and alfalfa are not nonrepetitive. In 2002, Alon et al. introduced a graph-theoretic generalisation of nonrepetitive sequences. They defined a vertex coloring of a graph to be nonrepetitive if the sequence of colors on any path in graph is nonrepetitive. Thue’s Theorem says that every path is nonrepetitively 3-colorable. Nonrepetitive graph coloring is a natural marriage of two major areas of combinatorial mathematics, combinatorics of words and graph coloring. In 2002, Alon et al. introduced both variants of nonrepetitive vertex or edge colorings of graphs but emphasized the edge coloring variant. Subsequent papers seem to have given more attention to the vertex coloring variant. Several authors have also studies nonrepetitive list coloring, facial nonrepetitive coloring and total nonrepetitive coloring. In this thesis, we study the nonrepetitive vertex coloring of graphs. The nonrepetitive chromatic number of a graph, denoted by (G) is the minimum number of colors in a nonrepetitive coloring of the vertices of the graph. Observe that every nonrepetitive coloring is proper, in the sense that adjacent vertices receive distinct colors. Moreover, a nonrepetitive coloring has no 2-colored P4 (a path on four vertices). A proper coloring with no 2-colored P4 is called a star coloring since each bichromatic subgraph is a star forest. The star chromatic number s(G) is the minimum number of colors in a star coloring of G. Thus (G) s(G) (G): Deciding whether a vertex coloring of a graph is nonrepetitive is coNP-complete. The most important problem in the field of nonrepetitive coloring is finding upper bounds on nonrepetitive chromatic number for various graph classes, including for the following graph classes. Cycles, trees, outerplanar graphs, graphs with bounded treewidth, graphs with bounded degree, planar graphs, graphs embeddable on a fixed surface, graphs excluding a fixed immersion (a graph G contains a graph H as an immersion if the vertices of H can be mapped to distinct vertices of G, and the edges of H can be mapped to pairwise edge‐disjoint paths in G, such that each edge vw of H is mapped to a path in G whose endpoints are the images of v and w), graphs excluding a fixed minor, graphs excluding a fixed topological minor (a graph G contains H as a topological minor if a subdivision of H is a subgraph of G), and graph subdivisions.
