Albertson conjectured that if graph G has chromatic number r, then the crossing number of G is at least that of the complete graph Kr. This conjecture in the case r = 5 is equivalent to the four color theorem. It was verified for r = 6 by Oporowski and Zhao. In this paper, we prove the conjecture for 7 ≤ r ≤ 12 using results of Dirac; Gallai; and Kostochka and Stiebitz that give lower bounds on...

The 4 Color Theorem (4CT) implies that every n-vertex planar graph has an independent set of size at least n4 ; this is best possible, as shown by the disjoint union of many copies of K4. In 1968, Erdős asked whether this bound on independence number could be proved more easily than the full 4CT. In 1976 Albertson showed (independently of the 4CT) that every n-vertex planar graph has an indepen...

The distinguishing number D(G) of a graph G is the least integer d such that there is a d-labeling of the vertices of G which is not preserved by any nontrivial automorphism. For a graph G let G be the rth power of G with respect to the Cartesian product. It is proved that D(G) = 2 for any connected graph G with at least 3 vertices and for any r ≥ 3. This confirms and strengthens a conjecture o...

In a graph G of maximum degree ∆ let γ denote the largest fraction of edges that can be ∆ edge-coloured. Albertson and Haas showed that γ ≥ 13 15 when G is cubic [1]. We show here that this result can be extended to graphs with maximum degree 3 with the exception of a graph on 5 vertices. Moreover, there are exactly two graphs with maximum degree 3 (one being obviously the Petersen graph) for w...

Let G be a simple graph with n vertices and list chromatic number χl(G) = χl. Suppose that 0 ≤ t ≤ χl and each vertex of G is assigned a list of t colors. Albertson, Grossman and Haas [1] conjectured that at least tn χl vertices of G can be colored from these lists. In this paper we find some new results in partial list coloring which help us to show that the conjecture is true for at least hal...

Let G be a graph with circular chromatic number χc(G) = k q . Given P ⊆ V (G) where each component of G[P ] is isomorphic to the circular clique Gk,q, suppose the vertices of P have been precoloured with a (k, q)-colouring. We examine under what conditions one can be assured the precolouring extends to the entire graph. We study sufficient conditions based on k ′ q − k q as well as the distance...

To cite: Albertson M, Jamous F, Groskreutz D. BMJ Case Rep Published online: [please include Day Month Year] doi:10.1136/ bcr-2014-208556 DESCRIPTION A 46-year-old male with a history of IV drug use (methamphetamine and heroin) presented with upper respiratory complaints. His chest X-ray (figure 1A, B) showed increased fullness in the area of the pulmonary artery. The successive chest CT (figur...

In this paper we define and study the distinguishing chromatic number, χD(G), of a graph G, building on the work of Albertson and Collins who studied the distinguishing number. We find χD(G) for various families of graphs and characterize those graphs with χD(G) = |V (G)|, and those trees with the maximum chromatic distingushing number for trees. We prove analogs of Brooks’ Theorem for both the...

Colour patterns in animals have long offered an opportunity to observe adaptive traits in natural populations. Colour plays myriad roles in interactions within and among species, from reproductive signalling to predator avoidance, leading to multiple targets of natural and sexual selection and opportunities for diversification. Understanding the genetic and developmental underpinnings of variat...