Distance graphs with maximum chromatic number
نویسندگان
چکیده
منابع مشابه
Distance graphs with maximum chromatic number
Let D be a finite set of integers. The distance graph G(D) has the set of integers as vertices and two vertices at distance d ∈ D are adjacent in G(D). A conjecture of Xuding Zhu states that if the chromatic number of G(D) achieves its maximum value |D| + 1 then the graph has a clique of order |D|. We prove that the chromatic number of a distance graph with D = {a, b, c, d} is five if and only ...
متن کاملUnit Distance Graphs with Ambiguous Chromatic Number
First László Székely and more recently Saharon Shelah and Alexander Soifer have presented examples of infinite graphs whose chromatic numbers depend on the axioms chosen for set theory. The existence of such graphs may be relevant to the Chromatic Number of the Plane problem. In this paper we construct a new class of graphs with ambiguous chromatic number. They are unit distance graphs with ver...
متن کاملThe locating-chromatic number for Halin graphs
Let G be a connected graph. Let f be a proper k -coloring of G and Π = (R_1, R_2, . . . , R_k) bean ordered partition of V (G) into color classes. For any vertex v of G, define the color code c_Π(v) of v with respect to Π to be a k -tuple (d(v, R_1), d(v, R_2), . . . , d(v, R_k)), where d(v, R_i) is the min{d(v, x)|x ∈ R_i}. If distinct vertices have distinct color codes, then we call f a locat...
متن کاملPacking Chromatic Number of Distance Graphs
The packing chromatic number (G) of a graph G is the smallest integer k such that vertices of G can be partitioned into disjoint classes X1; :::; Xk where vertices in Xi have pairwise distance greater than i. We study the packing chromatic number of in nite distance graphs G(Z; D), i.e. graphs with the set Z of integers as vertex set and in which two distinct vertices i; j 2 Z are adjacent if a...
متن کاملGraphs with chromatic number close to maximum degree
Let G be a color-critical graph with χ(G) ≥ Δ(G) = 2t + 1 ≥ 5 such that the subgraph of G induced by the vertices of degree 2t+1 has clique number at most t−1. We prove that then either t ≥ 3 and G = K2t+2 or t = 2 and G ∈ {K6, O5}, where O5 is a special graph with χ(O5) = 5 and |O5| = 9. This result for t ≥ 3 improves a case of a theorem by Rabern [9] and for t = 2 answers a question raised by...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 2008
ISSN: 0012-365X
DOI: 10.1016/j.disc.2007.07.061