The adaptable chromatic number of a multigraph G, denoted χa(G), is the smallest integer k such that every edge labeling of G from [k] = {1, 2, · · · , k} permits a vertex coloring of G from [k] such that no edge e = uv has c(e) = c(u) = c(v). Hell and Zhu proved that for any multigraph G with maximum degree ∆, the adaptable chromatic number is at most lp e(2∆− 1) m . We strengthen this to the ...

Hajj os theorem states that every graph with chromatic number at least k can be obtained from the complete graph K k by a sequence of simple operations such that every intermediate graph also has chromatic number at least k. Here, Hajj os theorem is extended in three slightly diierent ways to colorings and circular colorings of edge-weighted graphs. These extensions shed some new light on the H...

We consider infinite graphs. The distinguishing number D(G) of a graph G is the minimum number of colours in a vertex colouring of G that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by D′(G). We prove that D′(G) 6 D(G) + 1. For proper colourings, we study relevant invariants called the distinguishing chrom...

Let χl (G), χ ′ l (G), χ ′′ l (G), and 1(G) denote, respectively, the list chromatic number, the list chromatic index, the list total chromatic number, and the maximum degree of a non-trivial connected outerplane graph G. We prove the following results. (1) 2 ≤ χl (G) ≤ 3 and χl (G) = 2 if and only if G is bipartite with at most one cycle. (2) 1(G) ≤ χ ′ l (G) ≤ 1(G) + 1 and χ ′ l (G) = 1(G) + ...

Discrimination for equiluminant chromatic stimuli that vary in L- and M-cone excitation depends on the chromaticity difference between the test field and the surrounding area. The current study investigated the effect of the proximity in space and time of a surround to the test field on chromatic contrast discrimination. The experimental paradigm isolated spatial, temporal, and spatial-and-temp...

In this article, a novel structure of photonic crystal fiber with nearly zero ultra-flattened chromatic dispersion and ultra-low confinement loss is presented. By replacing the circular air-holes of two first rings with the elliptical air-holes, a fiber with outstanding features of chromatic dispersion and confinement loss is designed. The proposed structure is optimized for operating in a wide...

A graph is chromatic-index-critical if it cannot be edge-coloured with ∆ colours (with ∆ the maximal degree of the graph), and if the removal of any edge decreases its chromatic index. The Critical Graph Conjecture stated that any such graph has odd order. It has been proved false and the smallest known counterexample has order 18 [18, 31]. In this paper we show that there are no chromatic-inde...

A subspace arrangement defined by intersections of hyperplanes of the braid arrangement can be encoded by an edge colored hypergraph. It turns out that the characteristic polynomial of this type of subspace arrangement is given by a generalized chromatic polynomial of the associated edge colored hypergraph. The main result of this paper supplies a sufficient condition for the existence of non-t...