This work brings together ideas of mixing graph colourings, discrete homotopy, and precolouring extension. A particular focus is circular colourings. We prove that all the (k, q)-colourings of a graph G can be obtained by successively recolouring a single vertex provided k/q ≥ 2col(G) along the lines of Cereceda, van den Heuvel and Johnson’s result for kcolourings. We give various bounds for su...

Albertson and Berman conjectured that every planar graph has an induced forest on half of its vertices. The best known lower bound, due to Borodin, is that every planar graph has an induced forest on two fifths of its vertices. In a related result, Chartran and Kronk, proved that the vertices of every planar graph can be partitioned into three sets, each of which induce a forest. We show tighte...

Albertson and Berman [1] conjectured that every planar graph has an induced forest on half of its vertices; the current best result, due to Borodin [3], is an induced forest on two fifths of the vertices. We show that the Albertson-Berman conjecture holds, and is tight, for planar graphs of treewidth 3 (and, in fact, for any graph of treewidth at most 3). We also improve on Borodin’s bound for ...

Let G be a graph embedded on a surface Sε with Euler genus ε > 0, and let P ⊂ V (G) be a set of vertices mutually at distance at least 4 apart. Suppose all vertices of G have H(ε)-lists and the vertices of P are precolored, where H(ε) = ⌊ 7+ √ 24ε+1 2 ⌋ is the Heawood number. We show that the coloring of P extends to a list-coloring of G and that the distance bound of 4 is best possible. Our re...

Rennie on “A mechanistic model linking insect (Hydropsychidae) silk nets to incipient sediment motion in gravel-bedded streams” Lindsey K. Albertson, Leonard S. Sklar, and Bradley J. Cardinale Stroud Water Research Center, Avondale, Pennsylvania, USA, Earth and Climate Sciences, San Francisco State University, San Francisco, California, USA, School of Natural Resources and Environment, Universi...

We use graph homomorphisms and the chromatic properties of the Petersen graph to prove some inequalities on systems of sets. These inequalities are then used to find counterexamples to a conjecture of Albertson and Collins on the monotonicity of the chromatic difference sequence of a vertex-transitive graph.

Albertson [3] has defined the irregularity of a simple undirected graph G = (V,E) as irr(G) = ∑ uv∈E |dG(u)− dG(v)| , where dG(u) denotes the degree of a vertex u ∈ V . Recently, this graph invariant gained interest in the chemical graph theory, where it occured in some bounds on the first and the second Zagreb index, and was named the third Zagreb index [13]. For general graphs with n vertices...

Received Let G = (V; E) be a graph with n vertices, chromatic number (G) and list chromatic number`(G): Suppose each vertex of V (G) is assigned a list of t colors. Albertson, Grossman and Haas 1] conjectured that at least t ` (G) n vertices can be colored properly from these lists. Albertson et. al. 1] and Chappell 3] proved partial results concerning this conjecture. This paper presents algor...

We give examples of vertex-transitive graphs with non-monotonic chromatic difference sequences, disproving a conjecture of Albertson and Collins on the monotonicity of the chromatic difference sequence of vertex-transitive graphs, and answering a question of Zhou on the achievability of circulants.

We show that every plane graph with maximum face size four whose all faces of size four are vertex-disjoint is cyclically 5-colorable. This answers a question of Albertson whether graphs drawn in the plane with all crossings independent are 5-colorable.