Chromatic Vertex Folkman Numbers

For graph G and integers a1 \u3e · · · \u3e ar \u3e 2, we write G → (a1, · · · , ar) v if and only if for every r-coloring of the vertex set V (G) there exists a monochromatic Kai in G for some color i ∈ {1, · · · , r}. The vertex Folkman number Fv(a1, · · · , ar; s) is defined as the smallest integer n for which there exists a Ks-free graph G of order n such that G → (a1, · · · , ar) v . It is well known that if G → (a1, · · · , ar) v then χ(G) \u3e m, where m = 1+Pr i=1(ai−1). In this paper we study such Folkman graphs G with chromatic number χ(G) = m, which leads to a new concept of chromatic Folkman numbers. We prove constructively some existential results, among others that for all r, s \u3e 2 there exist Ks+1-free graphs G such that G → (s, · · ·r , s) v and G has the smallest possible chromatic number r(s − 1) + 1 with respect to this property. Among others we conjecture that for every s \u3e 2 there exists a Ks+1-free graph G on Fv(s, s; s + 1) vertices with χ(G) = 2s − 1 and G → (s, s) v

Injective coloring of product graphs

The problem of injective coloring in graphs can be revisited through two different approaches: coloring the two-step graphs and vertex partitioning of graphs into open packing sets, each of which is equivalent to the injective coloring problem itself. Taking these facts into account, we observe that the injective coloring lies between graph coloring and domination theory. We make use of these three points of view in this paper so as to investigate the injective coloring of some well-known graph products. We bound the injective chromatic number of direct and lexicographic product graphs from below and above. In particular, we completely determine this parameter for the direct product of two cycles. We also give a closed formula for the corona product of two graphs

On Robust Colorings of Hamming-Distance Graphs

Hq(n, d) is defined as the graph with vertex set Znq and where two vertices are adjacent if their Hamming distance is at least d. The chromatic number of these graphs is presented for various sets of parameters (q, n, d). For the 4-colorings of the graphs H2(n, n − 1) a notion of robustness is introduced. It is based on the tolerance of swapping colors along an edge without destroying properness of the coloring. An explicit description of the maximally robust 4-colorings of H2 (n, n − 1) is presented

Some properties on the lexicographic product of graphs obtained by monogenic semigroups

In (Das et al. in J. Inequal. Appl. 2013:44, 2013), a new graph Gamma (S-M) on monogenic semigroups S-M (with zero) having elements {0, x, x(2), x(3),..., x(n)} was recently defined. The vertices are the non-zero elements x, x(2), x(3),..., x(n) and, for 1 <= i, j <= n, any two distinct vertices x(i) and x(j) are adjacent if x(i)x(j) = 0 in S-M. As a continuing study, in an unpublished work, some well-known indices (first Zagreb index, second Zagreb index, Randic index, geometric-arithmetic index, atom-bond connectivity index, Wiener index, Harary index, first and second Zagreb eccentricity indices, eccentric connectivity index, the degree distance) over Gamma (S-M) were investigated by the same authors of this paper. In the light of the above references, our main aim in this paper is to extend these studies to the lexicographic product over Gamma (S-M). In detail, we investigate the diameter, radius, girth, maximum and minimum degree, chromatic number, clique number and domination number for the lexicographic product of any two (not necessarily different) graphs Gamma (S-M(1)) and Gamma (S-M(2)).Selçuk ÜniversitesiSungkyunkwan University (BK21

Queue Layouts of Graph Products and Powers

A \emphk-queue layout of a graph G consists of a linear order σ of V(G), and a partition of E(G) into k sets, each of which contains no two edges that are nested in σ . This paper studies queue layouts of graph products and power