Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A totally odd H-subdivision means a subdivision of a graph H in which each edge of H corresponds to a path of odd length. Thus this concept is a generalization of a subdivision of H.#R##N##R##N#,  give a structure theorem for graphs without a fixed graph H as a totally odd subdivision. Namely, every graph with no totally odd H-subdivision has a tree-decomposition such that each piece is either#R##N##R##N#1. after deleting bounded number of vertices, an "almost" embedded graph into a bounded-genus surface, or#R##N##R##N#2. after deleting bounded number of vertices, a bipartite graph, or#R##N##R##N#3. after deleting bounded number of vertices, a graph with maximum degree at most f(|H|) for some function f of |H| (or a 6|H|-degenerate graph).#R##N##R##N#Moreover,  can obtain either a totally odd Kk-subdivision or such a tree-decomposition in polynomial time.#R##N##R##N# note that for minor-free graphs,  just need the first structure [37], while for odd-minor-free graphs,  need the first two structures [10]. For subdivision-free graphs,  need the first and the third structures [17, 29]. So our result can be vied as a combination of odd-minor-free graphs and subdivision-free graphs. The same conclusion of the structure theorem is true if  replace "totally odd" by "parity". Hence this generalizes the structure theorem for subdivision-free graphs [17, 29].#R##N##R##N# also consider coloring of graphs with no totally odd Kk-subdivision.  prove that any graph with no totally odd Kk-subdivision is 79k2/4-colorable. The bound on the chromatic number is essentially best possible since the correct order of the magnitude of the chromatic number even for graphs with no Kk-subdivision is Θ(k2). Our result improves the bound given by Thomassen [44]. Furthermore, it also generalizes the result by Bollobas and Thomason, and Komlos and Szemeredi, [6, 30] for graphs without Kk-subdivisions.#R##N##R##N#Finally  consider of coloring graphs with no totally odd Kk-subdivision in terms of an algorithmic view. Using our structure theorem,  give an approximation algorithm for coloring of a graph G without a fixed graph H as a totally odd subdivision, using 2χ(G) + 6(|H| − 1) colors, where χ(G) is chromatic number of G. The same conclusion is true if  replace "totally odd" by "parity".#R##N##R##N# point out that it is Unique-Game hard to obtain an O(k/log2 k)-approximation algorithm for graph-coloring of graphs with maximum degree at most k − 2 [2], and hence it is also Unique-Game hard to obtain an O(k/log2 k)-approximation algorithm for graph-coloring of graphs even without a Kk-subdivision (i.e, without the parity constraint). Thus our additive error Θ(k) is most likely best possible up to constant.
['Ken-ichi Kawarabayashi']
I am writing a research paper on a idea Totally odd subdivisions and parity subdivisions: structures and coloring
[{"from": "human", "value": "I am writing a research paper on a idea Totally odd subdivisions and parity subdivisions: structures and coloring"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A totally odd H-subdivision means a subdivision of a graph H in which each edge of H corresponds to a path of odd length. Thus this concept is a generalization of a subdivision of H.#R##N##R##N#,  give a structure theorem for graphs without a fixed graph H as a totally odd subdivision. Namely, every graph with no totally odd H-subdivision has a tree-decomposition such that each piece is either#R##N##R##N#1. after deleting bounded number of vertices, an \"almost\" embedded graph into a bounded-genus surface, or#R##N##R##N#2. after deleting bounded number of vertices, a bipartite graph, or#R##N##R##N#3. after deleting bounded number of vertices, a graph with maximum degree at most f(|H|) for some function f of |H| (or a 6|H|-degenerate graph).#R##N##R##N#Moreover,  can obtain either a totally odd Kk-subdivision or such a tree-decomposition in polynomial time.#R##N##R##N# note that for minor-free graphs,  just need the first structure [37], while for odd-minor-free graphs,  need the first two structures [10]. For subdivision-free graphs,  need the first and the third structures [17, 29]. So our result can be vied as a combination of odd-minor-free graphs and subdivision-free graphs. The same conclusion of the structure theorem is true if  replace \"totally odd\" by \"parity\". Hence this generalizes the structure theorem for subdivision-free graphs [17, 29].#R##N##R##N# also consider coloring of graphs with no totally odd Kk-subdivision.  prove that any graph with no totally odd Kk-subdivision is 79k2/4-colorable. The bound on the chromatic number is essentially best possible since the correct order of the magnitude of the chromatic number even for graphs with no Kk-subdivision is \u0398(k2). Our result improves the bound given by Thomassen [44]. Furthermore, it also generalizes the result by Bollobas and Thomason, and Komlos and Szemeredi, [6, 30] for graphs without Kk-subdivisions.#R##N##R##N#Finally  consider of coloring graphs with no totally odd Kk-subdivision in terms of an algorithmic view. Using our structure theorem,  give an approximation algorithm for coloring of a graph G without a fixed graph H as a totally odd subdivision, using 2\u03c7(G) + 6(|H| \u2212 1) colors, where \u03c7(G) is chromatic number of G. The same conclusion is true if  replace \"totally odd\" by \"parity\".#R##N##R##N# point out that it is Unique-Game hard to obtain an O(k/log2 k)-approximation algorithm for graph-coloring of graphs with maximum degree at most k \u2212 2 [2], and hence it is also Unique-Game hard to obtain an O(k/log2 k)-approximation algorithm for graph-coloring of graphs even without a Kk-subdivision (i.e, without the parity constraint). Thus our additive error \u0398(k) is most likely best possible up to constant."}]