In many applications of graph colouring the sizes of colour classes should not be too large. For example, in scheduling jobs (some of which could be performed at the same time), it is not good if the resulting schedule requires many jobs to occur at some specific time. An application of this type is discussed in [8]. A possible formalization of this restriction is the notion of equitable colour...

A proper vertex coloring of a graph G is equitable if the size of color classes differ by at most one. The equitable chromatic threshold of G, denoted by ∗Eq(G), is the smallest integer m such that G is equitably n-colorable for all n m. We prove that ∗Eq(G) = (G) if G is a non-bipartite planar graph with girth 26 and (G) 2 or G is a 2-connected outerplanar graph with girth 4. © 2007 Elsevier B...

Roman dominating function} on a digraph $D$ with vertex set $V(D)$ is a labeling$fcolon V(D)to {0, 1, 2}$such that every vertex with label $0$ has an in-neighbor with label $2$. A set ${f_1,f_2,ldots,f_d}$ ofRoman dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vin V(D)$,is called a {em Roman dominating family} (of functions) on $D$....

Wu, Zhang and Li [4] conjectured that the set of vertices of any simple graph G can be equitably partitioned into ⌈(∆(G) + 1)/2⌉ subsets so that each of them induces a forest of G. In this note, we prove this conjecture for graphs G with ∆(G) ≥ |G|/2.

To solve the access-balancing problem in distributed storage systems, we introduce a new combinatorial model, called MinVar model for fractional repetition (FR) codes. Since FR codes are based on graphs or set our is characterized by property that variance among sums of block-labels incident to fixed vertex minimized. This characterization different from Dau and Milenkovic's MaxMinSum while min...

Let G be a graph of size n with vertex set V (G) and edge set E(G). A ρlabeling of G is a one-to-one function h : V (G) → {0, 1, . . . , 2n} such that {min{|h(u)−h(v)|, 2n+1−|h(u)−h(v)|} : {u, v} ∈ E(G)} = {1, 2, . . . , n}. Such a labeling of G yields a cyclicG-decomposition ofK2n+1. It is known that 2-regular bipartite graphs, the vertex-disjoint union of C3’s, and the vertex-disjoint union o...

A vertex bimagic total labeling on a graph with v vertices and e edges is a one to one map taking the vertices and edges onto the integers 1, 2, 3, ...v + e with the property that the sum of the label on the vertex and the labels of its incident edges is one of the constants k1 or k2, independent of the choice of the vertex. In this paper we have discussed that bistar Bn,n are vertex bimagic to...

An injective map f : E(G) → {±1, ±2, · · · , ±q} is said to be an edge pair sum labeling of a graph G(p, q) if the induced vertex function f*: V (G) → Z − {0} defined by f*(v) = (Sigma e∈Ev) f (e) is one-one, where Ev denotes the set of edges in G that are incident with a vetex v and f*(V (G)) is either of the form {±k1, ±k2, · · · , ±kp/2} or {±k1, ±k2, · · · , ±k(p−1)/2} U {k(p+1)/2} accordin...

For any non-trivial abelian group A under addition a graph $G$ is said to be $A$-textit{magic}  if there exists a labeling $f:E(G) rightarrow A-{0}$ such that, the vertex labeling $f^+$  defined as $f^+(v) = sum f(uv)$ taken over all edges $uv$ incident at $v$ is a constant. An $A$-textit{magic} graph $G$ is said to be $Z_k$-magic graph if the group $A$ is $Z_k$  the group of integers modulo $k...