45 research outputs found

    Planar graphs are acyclically edge (Δ+5)(\Delta + 5)-colorable

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    An edge coloring of a graph GG is to color all the edges in the graph such that adjacent edges receive different colors. It is acyclic if each cycle in the graph receives at least three colors. Fiam{\v{c}}ik (1978) and Alon, Sudakov and Zaks (2001) conjectured that every simple graph with maximum degree Δ\Delta is acyclically edge (Δ+2)(\Delta + 2)-colorable -- the well-known acyclic edge coloring conjecture (AECC). Despite many major breakthroughs and minor improvements, the conjecture remains open even for planar graphs. In this paper, we prove that planar graphs are acyclically edge (Δ+5)(\Delta + 5)-colorable. Our proof has two main steps: Using discharging methods, we first show that every non-trivial planar graph must have one of the eight groups of well characterized local structures; and then acyclically edge color the graph using no more than Δ+5\Delta + 5 colors by an induction on the number of edges.Comment: Full version with 120 page

    Graph Partitioning With Input Restrictions

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    In this thesis we study the computational complexity of a number of graph partitioning problems under a variety of input restrictions. Predominantly, we research problems related to Colouring in the case where the input is limited to hereditary graph classes, graphs of bounded diameter or some combination of the two. In Chapter 2 we demonstrate the dramatic eect that restricting our input to hereditary graph classes can have on the complexity of a decision problem. To do this, we show extreme jumps in the complexity of three problems related to graph colouring between the class of all graphs and every other hereditary graph class. We then consider the problems Colouring and k-Colouring for Hfree graphs of bounded diameter in Chapter 3. A graph class is said to be H-free for some graph H if it contains no induced subgraph isomorphic to H. Similarly, G is said to be H-free for some set of graphs H, if it does not contain any graph in H as an induced subgraph. Here, the set H consists usually of a single cycle or tree but may also contain a number of cycles, for example we give results for graphs of bounded diameter and girth. Chapter 4 is dedicated to three variants of the Colouring problem, Acyclic Colouring, Star Colouring, and Injective Colouring. We give complete or almost complete dichotomies for each of these decision problems restricted to H-free graphs. In Chapter 5 we study these problems, along with three further variants of 3-Colouring, Independent Odd Cycle Transversal, Independent Feedback Vertex Set and Near-Bipartiteness, for H-free graphs of bounded diameter. Finally, Chapter 6 deals with a dierent variety of problems. We study the problems Disjoint Paths and Disjoint Connected Subgraphs for H-free graphs

    Equitable partition of planar graphs

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    An equitable kk-partition of a graph GG is a collection of induced subgraphs (G[V1],G[V2],…,G[Vk])(G[V_1],G[V_2],\ldots,G[V_k]) of GG such that (V1,V2,…,Vk)(V_1,V_2,\ldots,V_k) is a partition of V(G)V(G) and −1≤∣Vi∣−∣Vj∣≤1-1\le |V_i|-|V_j|\le 1 for all 1≤i<j≤k1\le i<j\le k. We prove that every planar graph admits an equitable 22-partition into 33-degenerate graphs, an equitable 33-partition into 22-degenerate graphs, and an equitable 33-partition into two forests and one graph.Comment: 12 pages; revised; accepted to Discrete Mat

    EUROCOMB 21 Book of extended abstracts

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    Acyclic, Star and Injective Colouring: A Complexity Picture for H-Free Graphs

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