The vector graph and the chromatic number of the plane, or how NOT to prove that χ(E2)>4
نویسندگان
چکیده
The chromatic number χ (E) of the plane is known to be some integer between 4 and 7, inclusive. We prove a limiting result that says, roughly, that one cannot increase the lower bound on χ (E) by pasting Moser spindles together—even countably many—by taking translations by Zlinear combinations of a certain set of vectors.
منابع مشابه
An Inertial Lower Bound for the Chromatic Number of a Graph
Let χ(G) and χf (G) denote the chromatic and fractional chromatic numbers of a graph G, and let (n+, n0, n−) denote the inertia of G. We prove that: 1 + max ( n+ n− , n− n+ ) 6 χ(G) and conjecture that 1 + max ( n+ n− , n− n+ ) 6 χf (G). We investigate extremal graphs for these bounds and demonstrate that this inertial bound is not a lower bound for the vector chromatic number. We conclude with...
متن کاملReed's Conjecture on hole expansions
In 1998, Reed conjectured that for any graph G, χ(G) ≤ ⌈ 2 ⌉, where χ(G), ω(G), and ∆(G) respectively denote the chromatic number, the clique number and the maximum degree of G. In this paper, we study this conjecture for some expansions of graphs, that is graphs obtained with the well known operation composition of graphs. We prove that Reed’s Conjecture holds for expansions of bipartite graph...
متن کاملCommon Neighborhood Graph
Let G be a simple graph with vertex set {v1, v2, … , vn}. The common neighborhood graph of G, denoted by con(G), is a graph with vertex set {v1, v2, … , vn}, in which two vertices are adjacent if and only if they have at least one common neighbor in the graph G. In this paper, we compute the common neighborhood of some composite graphs. In continue, we investigate the relation between hamiltoni...
متن کاملColouring of Graphs with at Most ( 2 − o ( 1 ) ) χ Vertices
Ohba has conjectured [9] that if the graph G has 2χ(G)+1 or fewer vertices then the list chromatic number and chromatic number of G are equal. In this paper we prove that this conjecture is asymptotically correct. More precisely we obtain that for any 0 < ǫ < 1, there exist an n0 = n0(ǫ) such that the list chromatic number of G equals its chromatic number, provided n0 ≤ |V (G)| ≤ (2 − ǫ)χ(G).
متن کاملInjective Edge Chromatic Index of a Graph
Three edges e1, e2 and e3 in a graph G are consecutive if they form a path (in this order) or a cycle of length three. An injective edge coloring of a graph G = (V,E) is a coloring c of the edges of G such that if e1, e2 and e3 are consecutive edges in G, then c(e1) 6= c(e3). The injective edge coloring number χ ′ i(G) is the minimum number of colors permitted in such a coloring. In this paper,...
متن کاملChromatic polynomials of some nanostars
Let G be a simple graph and (G,) denotes the number of proper vertex colourings of G with at most colours, which is for a fixed graph G , a polynomial in , which is called the chromatic polynomial of G . Using the chromatic polynomial of some specific graphs, we obtain the chromatic polynomials of some nanostars.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Australasian J. Combinatorics
دوره 66 شماره
صفحات -
تاریخ انتشار 2016