For a nontrivial connected graph $G$, let $c:V\left(G\right)\to \mathbb{N}$ be a vertex coloring of $G$ where adjacent vertices may be colored the same. For a vertex $v\in V\left(G\right)$, the neighborhood color set $\mathrm{NC}\left(v\right)$ is the set of colors of the neighbors of $v$. The coloring $c$ is called a set coloring if $\mathrm{NC}\left(u\right)\ne \mathrm{NC}\left(v\right)$ for every pair $u,v$ of adjacent vertices of $G$. The minimum number of colors required of such a coloring is called the set chromatic number ${\chi }_{\mathrm{s}}\left(G\right)$. We show that the decision variant of determining ${\chi }_{\mathrm{s}}\left(G\right)$ is NP-complete in the general case, and show that ${\chi }_{\mathrm{s}}\left(G\right)$ can be...

For a nontrivial connected graph $G$, let $cV\left(G\right)\to \mathbb{N}$ be a vertex coloring of $G$ where adjacent vertices may be colored the same. For a vertex $v$ of $G$, the neighborhood color set $\mathrm{NC}\left(v\right)$ is the set of colors of the neighbors of $v$. The coloring $c$ is called a set coloring if $\mathrm{NC}\left(u\right)\ne \mathrm{NC}\left(v\right)$ for every pair $u,v$ of adjacent vertices of $G$. The minimum number of colors required of such a coloring is called the set chromatic number ${\chi }_s\left(G\right)$. A study is made of the set chromatic number of the join $G+H$ of two graphs $G$ and $H$. Sharp lower and upper bounds...

Harary [10, p. 7] claims that Veblen [20, p. 2] first suggested to formalize simple graphs using simplicial complexes. We have developed basic terminology for simple graphs as at most 1-dimensional complexes. We formalize this new setting and then reprove Mycielski’s [12] construction resulting in a triangle-free graph with arbitrarily large chromatic number. A different formalization of similar material is in [15].

An L(2, 1)-coloring (or labeling) of a graph G is a vertex coloring f : V (G) → Z+ ∪ {0} such that |f(u) − f(v)| ≥ 2 for all edges uv of G, and |f(u)−f(v)| ≥ 1 if d(u, v) = 2, where d(u, v) is the distance between vertices u and v in G. The span of an L(2, 1)-coloring is the maximum color (or label) assigned by it. The span of a graph G is the smallest integer λ such that there exists an L(2, 1)-coloring of G with span λ. An L(2, 1)-coloring of a graph with span equal to the span of the graph is...

In this paper we consider two parameters generalization of the Fibonacci numbers and Pell numbers, named as the $(k,p)$-Fibonacci numbers. We give some new interpretations of these numbers. Moreover using these interpretations we prove some identities for the $(k,p)$-Fibonacci numbers.

Shannon-Vizing-type problems concerning the upper bound for a distance chromatic index of multigraphs G in terms of the maximum degree Δ(G) are studied. Conjectures generalizing those related to the strong chromatic index are presented. The chromatic d-index and chromatic d-number of paths, cycles, trees and some hypercubes are determined. Among hypercubes, however, the exact order of their growth is found.

Let Γ(R) be the zero divisor graph for a commutative ring with identity. The k-domination number and the 2-packing number of Γ(R), where R is an Artinian ring, are computed. k-dominating sets and 2-packing sets for the zero divisor graph of the ring of Gaussian integers modulo n, Γ(ℤₙ[i]), are constructed. The center, the median, the core, as well as the automorphism group of Γ(ℤₙ[i]) are determined. Perfect zero divisor graphs Γ(R) are investigated.

Let $R$ be a ring with identity and $M$ be a unitary left $R$-module. The co-intersection graph of proper submodules of $M$, denoted by $\Omega \left(M\right)$, is an undirected simple graph whose vertex set $V\left(\Omega \right)$ is a set of all nontrivial submodules of $M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $N+K\ne M$. We study the connectivity, the core and the clique number of $\Omega \left(M\right)$. Also, we provide some conditions on the module $M$, under which the clique number of $\Omega \left(M\right)$ is infinite and $\Omega \left(M\right)$ is a planar graph. Moreover, we give several...

It is proved that if G is multigraph with maximum degree 3, and every submultigraph of G has average degree at most 2(1/2) and is different from one forbidden configuration C⁺₄ with average degree exactly 2(1/2), then G is totally 4-choosable; that is, if every element (vertex or edge) of G is assigned a list of 4 colours, then every element can be coloured with a colour from its own list in such a way that no two adjacent or incident elements are coloured with the same colour. This shows that the...

Given a graph G = (V,E) and a (not necessarily proper) edge-coloring of G, we consider the complexity of finding a spanning tree of G with as many different colors as possible, and of finding one with as few different colors as possible. We show that the first problem is equivalent to finding a common independent set of maximum cardinality in two matroids, implying that there is a polynomial algorithm. We use the minimum dominating set problem to show that the second problem is NP-hard.