The Mycielski ideal Mk is defined to consist of all sets A ⊆ k such that {f ↾ X : f ∈ A} 6= k for all X ∈ [ω]0 . It will be shown that the covering numbers for these ideals are all equal. However, the covering numbers of the closely associated Ros lanowski ideals will be shown to be consistently different.

The Annual Reports of the Commissioner of Indian Affairs, for the years 1824 through 1920, described the interactions between the American Indian tribes and the federal government. J. A. Jones (Jones, J. A. (1955). Key to the annual reports of the Commissioner of Indian Affairs. Ethnohistory, 2, 58–64) provided a key to these documents found in the United States Congressional Serial Set, but th...

The Wiener number of a graph G is defined as 1 2 ∑ u,v∈V (G) d(u, v), d the distance function on G. The Wiener number has important applications in chemistry. We determine a formula for the Wiener number of an important graph family, namely, the Mycielskians μ(G) of graphs G. Using this, we show that for k ≥ 1, W (μ(S n)) ≤ W (μ(T k n )) ≤ W (μ(P k n )), where Sn, Tn and Pn denote a star, a gen...

Erratum: Identification and corrections of the existing mistakes in the paper On the total graph of Mycielski graphs, central graphs and their covering numbers, Discuss. Math. Graph Theory 33 (2013) 361–371.

This paper gives a sufficient condition for a graph G to have its circular chromatic number equal its chromatic number. By using this result, we prove that for any integer t ≥ 1, there exists an integer n such that for all k ≥ n χc(M (Kk)) = χ(M (Kk)).

A set $S$ of vertices is a determining for graph $G$ if every automorphism uniquely determined by its action on $S$. The size smallest called number, $Det(G)$. said to be $d$-distinguishable there coloring the with $d$ colors so that only trivial preserves color classes. such distinguishing $Dist(G)$. If $Dist(G) = 2$, cost 2-distinguishing, $\rho(G)$, class over all 2-distinguishing colorings ...

Domination game is a played on finite, undirected graph G, between two players Dominator and Staller. During the game, alternately choose vertices of G such that each chosen vertex dominates at least one new not dominated by previously vertices. The aim to finish as early possible while Staller delay process much possible. domination number γg(G) total moves in when starts both play optimally. ...

The local chromatic number of a graph was introduced in [12]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and...

Let ω(G) and χ(G) be the clique number and the chromatic number of a graph G. Mycielski [11] presented a construction that for any n creates a graph Mn which is triangle-free (ω(G) = 2) with χ(G) > n. The starting point is the complete graph of two vertices (K2). M(n+1) is obtained from Mn through the operation μ(G) called the Mycielskian of a graph G. We first define the operation μ(G) and the...

A k-fold coloring of a graph is a function that assigns to each vertex a set of k colors, so that the color sets assigned to adjacent Contract grant sponsor: NSFC; Contract grant number: 10671033 (to W.L.); Contract grant sponsor: National Science Foundation; Contract grant number: DMS 0302456 (to D.D.L.); Contract grant sponsor: National Science Council; Contract grant number: NSC95-2115-M-110...