The unit distance graph R is the graph with vertex set R2 in which two vertices (points in the plane) are adjacent if and only if they are at Euclidean distance 1. We prove that the circular chromatic number of R is at least 4, thus “tightening” the known lower bound on the chromatic number of R.

A subset D of vertices of a graph G is a dominating set if for each u ∈ V (G) \ D, u is adjacent to somevertex v ∈ D. The domination number, γ(G) ofG, is the minimum cardinality of a dominating set of G. A setD ⊆ V (G) is a total dominating set if for eachu ∈ V (G), u is adjacent to some vertex v ∈ D. Thetotal domination number, γt (G) of G, is theminimum cardinality of a total dominating set o...

In 1960 Berge came up with the concept of perfect graphs, and in doing so, conjectured some characteristics about them. A perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph [2]. Two conjectures are now known as the Perfect Graph Theorem and the Strong Perfect Graph Theorem. Both of these theorems make deteminin...

A vertex-distinguishing coloring of a graph G consists in an edge or a vertex coloring (not necessarily proper) of G leading to a labeling of the vertices of G, where all the vertices are distinguished by their labels. There are several possible rules for both the coloring and the labeling. For instance, in a set irregular edge coloring [5], the label of a vertex is the union of the colors of i...

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $n$. A $k$-vertex weightings of a graph $G$ is a mapping $w: V(G) to {1, ldots, k}$. A $k$-vertex weighting induces an edge labeling $f_w: E(G) to N$ such that $f_w(uv)=w(u)+w(v)$. Such a labeling is called an {it edge-coloring k-vertex weightings} if $f_{w}(e)not= f_{w}(echr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39'...

Given a set D of positive integers, the distance graph G(Z , D) has all integers as vertices, and two vertices are adjacent if and only if their difference is in D; that is, the vertex set is Z and the edge set is {uv : |u − v| ∈ D}. We call D the distance set. This paper studies chromatic and circular chromatic numbers of some distance graphs with certain distance sets. The circular chromatic ...

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by χ′a(G). Sierpinski graphs S(n, 3) are the graphs of the Tower of Hanoi with n disks, while Sierpinski gasket graphs Sn are the graphs naturally defined ...

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). A graph G is called fully subdivided if it is obtained from another graph H by replacing every edge by a path of length at least two. Fully subdi...

The purpose of this note is to improve Hind's theorem for graphs with large chromatic number by essentially reducing the power of / from | to | + e (the exact statement of our result is given in equations (4)-(6) below). Our proof uses a lemma which in words states that, if we assign to each vertex x in a fc-chromatic r-edge-chromatic multigraph a colour/(x) from {1,2,. . . ,r}, then there exis...

We address the chromatic number of a class of Euclidean distance graphs having vertex set Q, the 3-dimensional rational space. It is shown that if s is any positive integer with no prime factors congruent to 2 (mod 3), and G is the graph with vertex set Q where any two vertices are adjacent if and only if they are distance √ 2s apart, then G has chromatic number 4. Along the way, we obtain a fe...