We consider the Chromatic Sum Problem on bipartite graphs which appears to be much harder than the classical Chromatic Number Problem. We prove that the Chromatic Sum Problem is NP-complete on planar bipartite graphs with ∆ ≤ 5, but polynomial on bipartite graphs with ∆ ≤ 3, for which we construct an O(n)-time algorithm. Hence, we tighten the borderline of intractability for this problem on bip...

The mod sum number p( G) of a connected graph G is the minimum number of isolated vertices required to transform G into a mod sum graph. It is known that the mod sum number is greater than zero for wheels, Wn, when n > 4 and for the complete graphs, Kn when n 2: 2. In this paper we show that p( Hm,n) > 0 for n > m ;::: 3. Vie show further that P(K2) = P(K3) = 1 while p(Kn) = n for n ;::: 4. We ...

Batch scheduling of conflicting jobs is modeled by batch coloring of a graph. Given an undirected graph and the number of colors required by each vertex, we need to find a proper batch coloring of the graph, i.e., partition the vertices to batches which are independent sets, and to assign to each batch a contiguous set of colors, whose size equals to the maximum color requirement of any vertex ...

Let R be a commutative ring with identity. An ideal I of a ring R is called an annihilating ideal if there exists r ∈ R \ {0} such that Ir = (0) and an ideal I of R is called an essential ideal if I has non-zero intersection with every other non-zero ideal of R. The sum-annihilating essential ideal graph of R, denoted by AER, is a graph whose vertex set is the set of all non-zero annihilating i...

We consider a general class of scheduling problems where a set of dependent jobs needs to be scheduled (preemptively or nonpreemptively) on a set of machines so as to minimize the weighted sum of completion times. The dependencies among the jobs are formed as an arbitrary conflict graph. An input to our problems can be modeled as an instance of the sum multicoloring (SMC) problem: Given a graph...

In this paper, we study the square of generalized Hamming graphs by properties abelian groups, and characterize some isomorphisms between non-complete extended p-sum complete graphs. As applications, determine eigenvalues

For a fixed positive ϵ, we show the existence of constant Cϵ with following property: Given ±1-edge-labeling c:E(Kn)→{−1,1} complete graph Kn c(E(Kn))=0, and spanning forest F maximum degree Δ, one can determine in polynomial time an isomorphic copy F′ |c(E(F′))|≤34+ϵΔ+Cϵ. Our approach is based on method conditional expectation.

We introduce a variant of the well-studied sum choice number of graphs, which we call the interactive sum choice number. In this variant, we request colours to be added to the vertices’ colour-lists one at a time, and so we are able to make use of information about the colours assigned so far to determine our future choices. The interactive sum choice number cannot exceed the sum choice number ...

The problem of minimum color sum of a graph is to color the vertices of the graph such that the sum (average) of all assigned colors is minimum. Recently, in [BBH+96], it was shown that in general graphs this problem cannot be approximated within n1 , for any > 0, unless NP = ZPP . In the same paper, a 9=8-approximation algorithm was presented for bipartite graphs. The hardness question for thi...

the vertex-edge wiener index of a simple connected graph g is defined as the sum of distances between vertices and edges of g. two possible distances d_1(u,e|g) and d_2(u,e|g) between a vertex u and an edge e of g were considered in the literature and according to them, the corresponding vertex-edge wiener indices w_{ve_1}(g) and w_{ve_2}(g) were introduced. in this paper, we present exact form...