A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number χ CF (G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N [v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N (v), for which vertex v is not a member of its neighborhood.For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K k+1 as a minor, then χ CF (G) ≤ k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. In addition, we give a complete characterization of the algorithmic/computational complexity of conflict-free coloring. It is NP-complete to decide whether a planar graph has a conflict-free coloring with one color, while for outerplanar graphs, this can be decided in polynomial time. Furthermore, it is NP-complete to decide whether a planar graph has a conflict-free coloring with two colors, while for outerplanar graphs, two colors always suffice. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs.For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k ∈ {1, 2, 3}, it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general planar graph has a conflict-free coloring with at most eight colors.2. It is NP-complete to decide whether a planar graph has a conflict-free coloring with one color.For outerplanar graphs, this question can be decided in polynomial time.

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number χ CF (G) (the smallest k for which conflict-free k-colorings exist), with a focus on planar graphs.For general graphs, we prove the conflict-free variant of the famous Hadwiger Conjecture: If G does not contain K k+1 as a minor, then χ CF (G) ≤ k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. In addition, we give a complete characterization of the algorithmic/computational complexity of conflict-free coloring. It is NP-complete to decide whether a planar graph has a conflict-free coloring with one color, while for outerplanar graphs, this can be decided in polynomial time. Furthermore, it is NP-complete to decide whether a planar graph has a conflict-free coloring with two colors, while for outerplanar graphs, two colors always suffice. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs.

We consider problems in which a simple path of fixed length, in an undirected graph, is to be shifted from a start position to a goal position by moves that add an edge to either end of the path and remove an edge from the other end. We show that this problem may be solved in linear time in trees, and is fixed-parameter tractable when parameterized either by the cyclomatic number of the input graph or by the length of the path. However, it is PSPACE-complete for paths of unbounded length in graphs of bounded bandwidth.

Abstract:We prove the computational intractability of rotating and placing n square tiles into a 1 × n array such that adjacent tiles are compatible-either equal edge colors, as in edge-matching puzzles, or matching tab/pocket shapes, as in jigsaw puzzles. Beyond basic NP-hardness, we prove that it is NP-hard even to approximately maximize the number of placed tiles (allowing blanks), while satisfying the compatibility constraint between nonblank tiles, within a factor of 0.9999999702 (On the other hand, there is an easy 1 2 -approximation). This is the first (correct) proof of inapproximability for edge-matching and jigsaw puzzles. Along the way, we prove NP-hardness of distinguishing, for a directed graph on n nodes, between having a Hamiltonian path (length n − 1) and having at most 0.999999284(n − 1) edges that form a vertex-disjoint union of paths. We use this gap hardness and gap-preserving reductions to establish similar gap hardness for 1 × n jigsaw and edge-matching puzzles.

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