Some generalizations of theorems on vertex coloring

The Vertex Coloring Problem and its generalizations

Generalizations of Some Zero-sum Theorems

For a finite abelian group G and a finite subset A ⊆ Z, the Davenport constant of G with weight A, denoted by DA(G), is defined to be the smallest positive integer k such that for any sequence (x1, . . . , xk) of k elements in G there exists a non-empty subsequence (xj1 , . . . , xjr) and a1, . . . , ar ∈ A such that ∑r i=1 ai xji = 0. To avoid trivial cases, one assumes that the weight set A d...

Some New Generalizations of Karapinar’s Theorems

In-depth study of fixed point theory is to solve the existence of the equation Tx = x (or x ∈ T (x)), where T is a self-map or a non-self-map. However, as we know, the equation Tx = x (or x ∈ T (x)) is not necessarily to have a solution. So, we turn to explore the best approximation of the existence of solutions. In 2003, Kirk-Srinavasan-Veeramani [1] introduced cyclic maps and best proximity p...

Edge-coloring Vertex-weightings of Graphs

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $n$. A $k$-vertex weightings of a graph $G$ is a mapping $w: V(G) to {1, ldots, k}$. A $k$-vertex weighting induces an edge labeling $f_w: E(G) to N$ such that $f_w(uv)=w(u)+w(v)$. Such a labeling is called an {it edge-coloring k-vertex weightings} if $f_{w}(e)not= f_{w}(echr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39'...

On Some Generalizations of Fermat's, Lucas's and Wilson's Theorems

“Never underestimate a theorem that counts something!” – or so says J. Fraleigh in his classic text [2]. Indeed, in [1] and [4], the authors derive Fermat’s (little), Lucas’s and Wilson’s theorems, among other results, all from a single combinatorial lemma. This lemma can be derived by applying Burnside’s theorem to an action by a cyclic group of prime order. In this note, we generalize this le...