In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A k-stack (resp...

A queue layout of a graph consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its queue-number. A three-dimensional (straight- line grid) drawing of a graph represents the vertices by points in Z and the edges by non-crossing line-segments. This paper contributes three main results: (1) It is proved that the minimum volume of a certain type of three-dimensional drawing of a graph G is closely related to the queue-number of G. In particular, if G is an n-vertex member of a proper minor-closed family of graphs (such as a planar graph), then G has a O(1) O(1) O(n) drawing if and only if G has O(1) queue-number.

A proper coloring of the vertices of a graph is called a star coloring if every two color classes induce a star forest. Star colorings are a strengthening of acyclic colorings, i.e., proper colorings in which every two color classes induce a forest. We show that

Let G be a graph with chromatic number χ(G). A vertex colouring of G is acyclic if each bichromatic subgraph is a forest. A star colouring of G is an acyclic colouring in which each bichromatic subgraph is a star forest. Let χa(G) and χs(G) denote the acyclic and star chromatic numbers of G. This paper investigates acyclic and star colourings of subdivisions. Let G ′ be the graph obtained from G by subdividing each edge once. We prove that acyclic (respectively, star) colourings of G ′ correspond to vertex partitions of G in which each subgraph has small arboricity (chromatic index). It follows that χa(G ′), χs(G ′ ) and χ(G) are tied, in the sense that each is bounded by a function of the other. Moreover the binding functions that we establish are all tight. The oriented chromatic number − → χ (G) of an (undirected) graph G is the maximum, taken over all orientations D of G, of the minimum number of colours in a vertex colouring of D such that between any two colour classes, all edges have the same direction. We prove that − → χ (G ′ ) = χ(G) whenever χ(G) ≥ 9.

. A graph is 1-planar if it can be drawn on the plane in such a way that every edge crosses at most one other edge. We prove that the acyclic chromatic number of every 1-planar graph is at most 20. Keywords. Acyclic colouring, planar graphs, 1-planar graphs. 1

We obtain bounds for the coloring numbers of products of trees for three closely related types of colorings: acyclic, distance 2, and L(2, 1).

In this paper, we show that the dimension of the adjacency poset of a planar graph is at most 8. From below, we show that there is a planar graph whose adjacency poset has dimension 5. We then show that the dimension of the adjacency poset of an outerplanar graph is at most 5. From below, we show that there is an outerplanar graph whose adjacency poset has dimension 4. We also show that the dimension of the adjacency poset of a planar bipartite graph is at most 4. This result is best possible. More generally, the dimension of the adjacency poset of a graph is bounded as a function of its genus and so is the dimension of the vertex-face poset of such a graph.

We relate acyclic (and star) chromatic number of a graph to the chromatic number of its minors and as a consequence we show that the set of all triangle free planar graphs is is homomorphism bounded by a triangle free graph. This solves a problem posed in [14] and completes results of [13] where a similar result has been proved for Kk -free graphs for k = 4; 5. It also

Abstract. Let F be a family of connected bipartite graphs, each with at least three vertices. A proper vertex colouring of a graph G with no bichromatic subgraph in F is F-free. The F-free chromatic number χ(G, F) of a graph G is the minimum number of colours in an F-free colouring of G. For appropriate choices of F, several well-known types of colourings fit into this framework, including acyclic colourings, star colourings, and distance-2 colourings. This paper studies F-free colourings of the cartesian product of graphs. Let H be the cartesian product of the graphs G1, G2,..., Gd. Our main result establishes an upper bound on the F-free chromatic number of H in terms of the maximum F-free chromatic number of the Gi and the following number-theoretic concept. A set S of natural numbers is k-multiplicative Sidon if ax = by implies a = b and x = y whenever x,y ∈ S and 1 ≤ a, b ≤ k. Suppose that χ(Gi, F) ≤ k and S is a k-multiplicative Sidon set of cardinality d. We prove that χ(H, F) ≤ 1+2k·max S. We then prove that the maximum density of a k-multiplicative Sidon set is Θ(1/log k). It follows that χ(H, F) ≤ O(dk log k). We illustrate the method with numerous examples, some of which generalise or improve upon existing results in the literature. 1.

A track layout of a graph consists of a vertex colouring, an edge colouring, and a total order of each vertex colour class such that between each pair of vertex colour classes, there is no monochromatic pair of crossing edges. This paper studies track layouts and their applications to other models of graph layout. In particular, we improve on the results of Enomoto and Miyauchi [SIAM J. Discrete Math., 1999] regarding stack layouts of subdivisions, and give analogous results for queue layouts. We solve open problems due to Felsner, Wismath, and Liotta [Proc. Graph Drawing, 2001] and Pach, Thiele, and Toth [Proc. Graph Drawing, 1997] concerning three-dimensional straight-line grid drawings. We initiate the study of three-dimensional polyline grid drawings and establish volume bounds within a logarithmic factor of optimal.