On some properties of suboptimal colorings of graphs

Abstract

Starting from the trivial observation that, in any optimal coloring of a graph, there always exists a node v such that its neighborhood N(v) contains all colors, we examine related properties in suboptimal colorings (i.e., those using more than (G) colors, where (G) is the chromatic number). In particular, we show that, in any ((G) ؉ p)coloring of G, there is a node v such that its generalized neighborhood N q (v) with q ‫؍‬ max{2p ؊ 1, 2} contains (G) colors for p ≥ 1. Additional properties of ((G) ؉ p)colorings are also given.

Locally restricted colorings

“…We shall now establish a bound on ( , p) which will strengthen a result given in [2]. This can be formulated by stating ( , p) p/2 + 1 for all and p. Notice that for p = 0, Theorem 8 reduces to Proposition 3.…”

“…By construction, we have K 1 (v i,{j,k} )={i, j, k}. In [2], we have shown that (pentaK 3,3 )= 4. We shall now prove that pentaK 3,3 has a minimum number of vertices.…”

“…If (2) and the existence of the path P, and Q a(v) (v) ⊆ Q r(u) (u) by (5), and this implies (7) and hence (8).…”

“…This can be formulated by stating ( , p) p/2 + 1 for all and p. Notice that for p = 0, Theorem 8 reduces to Proposition 3. In [2] the Property was shown for N q [v] with q = max{2p − 1, 2} instead of q = p/2 + 1.…”

“…In [2], some basic properties of k-colorings with k > have been derived, and a few questions were raised with respect to such colorings.…”