157 research outputs found

    Track Layouts of Graphs

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    A \emph{(k,t)(k,t)-track layout} of a graph GG consists of a (proper) vertex tt-colouring of GG, a total order of each vertex colour class, and a (non-proper) edge kk-colouring such that between each pair of colour classes no two monochromatic edges cross. This structure has recently arisen in the study of three-dimensional graph drawings. This paper presents the beginnings of a theory of track layouts. First we determine the maximum number of edges in a (k,t)(k,t)-track layout, and show how to colour the edges given fixed linear orderings of the vertex colour classes. We then describe methods for the manipulation of track layouts. For example, we show how to decrease the number of edge colours in a track layout at the expense of increasing the number of tracks, and vice versa. We then study the relationship between track layouts and other models of graph layout, namely stack and queue layouts, and geometric thickness. One of our principle results is that the queue-number and track-number of a graph are tied, in the sense that one is bounded by a function of the other. As corollaries we prove that acyclic chromatic number is bounded by both queue-number and stack-number. Finally we consider track layouts of planar graphs. While it is an open problem whether planar graphs have bounded track-number, we prove bounds on the track-number of outerplanar graphs, and give the best known lower bound on the track-number of planar graphs.Comment: The paper is submitted for publication. Preliminary draft appeared as Technical Report TR-2003-07, School of Computer Science, Carleton University, Ottawa, Canad

    The Relaxed Game Chromatic Index of \u3cem\u3ek\u3c/em\u3e-Degenerate Graphs

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    The (r, d)-relaxed coloring game is a two-player game played on the vertex set of a graph G. We consider a natural analogue to this game on the edge set of G called the (r, d)-relaxed edge-coloring game. We consider this game on trees and more generally, on k-degenerate graphs. We show that if G is k-degenerate with ∆(G) = ∆, then the first player, Alice, has a winning strategy for this game with r = ∆+k−1 and d≥2k2 + 4k

    Colorings of oriented planar graphs avoiding a monochromatic subgraph

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    For a fixed simple digraph FF and a given simple digraph DD, an FF-free kk-coloring of DD is a vertex-coloring in which no induced copy of FF in DD is monochromatic. We study the complexity of deciding for fixed FF and kk whether a given simple digraph admits an FF-free kk-coloring. Our main focus is on the restriction of the problem to planar input digraphs, where it is only interesting to study the cases k∈{2,3}k \in \{2,3\}. From known results it follows that for every fixed digraph FF whose underlying graph is not a forest, every planar digraph DD admits an FF-free 22-coloring, and that for every fixed digraph FF with Δ(F)≥3\Delta(F) \ge 3, every oriented planar graph DD admits an FF-free 33-coloring. We show in contrast, that - if FF is an orientation of a path of length at least 22, then it is NP-hard to decide whether an acyclic and planar input digraph DD admits an FF-free 22-coloring. - if FF is an orientation of a path of length at least 11, then it is NP-hard to decide whether an acyclic and planar input digraph DD admits an FF-free 33-coloring
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