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'...

a vertex irregular total k-labeling of a graph g with vertex set v and edge set e is an assignment of positive integer labels {1, 2, ..., k} to both vertices and edges so that the weights calculated at vertices are distinct. the total vertex irregularity strength of g, denoted by tvs(g)is the minimum value of the largest label k over all such irregular assignment. in this paper, we study the to...

The spectra of previously constructed auxiliary matrices for the six-vertex model at roots of unity are investigated for spin-chains of even and odd length. The two cases show remarkable differences. In particular, it is shown that for even roots of unity and an odd number of sites the eigenvalues contain two linear independent solutions to Baxter’s TQ-equation corresponding to the Bethe ansatz...

Let G = (V, E) be a graph with v vertices and e edges. An (a, d)-vertex-antimagic total labeling is a bijection λ from V (G) ∪ E(G) to the set of consecutive integers 1, 2, . . . , v + e, such that the weights of the vertices form an arithmetic progression with the initial term a and common difference d. If λ(V (G)) = {1, 2, . . . , v} then we call the labeling a super (a, d)-vertex-antimagic t...

This paper presents an interactive modeling system of uniform polyhedra using the isomorphic graphs. Especially, Kepler-Poinsot solids are formed by modifications of dodecahedron and icosahedron. Keywords—Kepler-Poinsot solid, Shape modeling, Polyhedral graph, Graph drawing. I. INTRUDUCTION EPLER-POINSOT solids are subset of uniform polyhedra, which include 5 regular polyhedra (Platonic soilids...

Assume that we have m identical graphs where the graphs consists of paths with k vertices where k is a positive integer. In this paper, we discuss certain labelling of the m graphs called c-Erdösian for some positive integers c. We regard labellings of the vertices of the graphs by positive integers, which induce the edge labels for the paths as the sum of the two incident vertex labels. They h...

Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no (k−ǫ) poly(n) time algorithm for deciding if an n-vertex graph G with pathwidth pw(G) admits a proper vertex coloring with k colors unless the Strong Exponential Time Hypothesis (SETH) is false. We show here that nevertheless, when k > ⌊∆/2⌋+1, where ∆ is the maximum degree in the graph G, there is a better algorithm, at least when...

Cai and Yang initiated the systematic parameterized complexity study of the following set of problems around Eulerian graphs. For a given graph G and integer k, the task is to decide if G contains a (connected) subgraph with k vertices (edges) with all vertices of even (odd) degrees. They succeed to establish the parameterized complexity of all cases except two, when we ask about a connected k-...

An assignment of integer numbers to the vertices of a given graph under certain conditions is referred to as a graph labeling. The assignment of labels from the set {0,1,2,...,2 1} q to the vertices of G (with ( ) G n V vertices and ( ) q E G edges) such that, when each edge has assigned a label defined by the absolute difference of its end-points, the resulting edge labels are 1,3 ,2 1 q is re...