Chromatic Sums Research Articles

Finding the chromatic sums of graphs using a D-Wave quantum computer

In this paper we demonstrate how to solve the chromatic sum problem using a D-Wave quantum computer. Starting from a BIP (binary integer programming) formulation, we develop a QUBO (quadratic unconstrained binary optimization) formulation of the chromatic sum problem, which is acceptable to a D-Wave quantum computer. Our construction requires nk qubits for a graph of n vertices and upper bound of k colors. Further, we present the experimental results obtained by running several QUBOs on a D-Wave quantum computer.

Certain chromatic sums of some cycle-related graph classes

Let [Formula: see text] be a certain type of proper [Formula: see text]-coloring of a given graph [Formula: see text] and [Formula: see text] denote the number of times a particular color [Formula: see text] is assigned to the vertices of [Formula: see text]. Then, the coloring sum of a given graph [Formula: see text] with respect to the coloring [Formula: see text], denoted by [Formula: see text] is defined to be [Formula: see text]. The coloring sums such as [Formula: see text]-chromatic sum, [Formula: see text]-chromatic sum, [Formula: see text]-chromatic sum, [Formula: see text]-chromatic sum, etc. are some of these types of coloring sums that have been studied recently. Motivated by these studies on certain chromatic sums of graphs, in this paper, we study certain chromatic sums for some standard cycle-related graphs.

Chromatic Sums for Colorings Avoiding Monochromatic Subgraphs

Abstract Given graphs G and H, a vertex coloring c : V ( G ) → N is an H-free coloring of G if no color class contains a subgraph isomorphic to H. The H-free chromatic number of G, χ ( H , G ) , is the minimum number of colors in an H-free coloring of G. The H-free chromatic sum of G , Σ ( H , G ) , is the minimum value achieved by summing the vertex colors of each H-free coloring of G. We provide a general bound for Σ ( H , G ) , discuss the computational complexity of finding this parameter for different choices of H, and prove an exact formulas for some graphs G. For every integer k and for every graph H, we construct families of graphs, G k with the property that k more colors than χ ( H , G ) are required to realize Σ ( H , G ) for H-free colorings. More complexity results and constructions of graphs requiring extra colors are given for planar and outerplanar graphs.

Chromatic sums of biloopless nonseparable near-triangulations on the projective plane

In this paper, the chromatic sum functions of rooted biloopless nonseparable near-triangulations on the sphere and the projective plane are studied. The chromatic sum function equations of such maps are obtained. From the chromatic sum equations of such maps, the enumerating function equations of such maps are derived. An asymptotic evaluation and some explicit expression of enumerating functions are also derived.

Chromatic sums of nonseparable near-triangulations on the projective plane

In this paper, we study the chromatic sum functions of rooted nonseparable near-triangulations on the sphere and the projective plane. The chromatic sum function equations of such maps are obtained. From the chromatic sum equations of such maps, the enumerating function equations of such maps are derived. Applying chromatic sum theory, the enumerating problem of different sorts maps can be studied, and a new method of enumeration can be obtained. Moreover, an asymptotic evaluation and some explicit expression of enumerating functions are also derived.

Chromatic sums of general maps on the sphere and the projective plane

In this paper, we study the chromatic sum functions of rooted general maps on the sphere and the projective plane. The chromatic sum function equations of such maps are obtained. From the chromatic sum equations of such maps, the enumerating function equations of rooted loopless maps, bipartite maps and Eulerian maps are also derived. Moreover, some explicit expressions of enumerating functions are also derived.

Chromatic sums of 2-edge-connected maps on the plane

This paper provides the chromatic sum function equation of rooted 2-edge-connected maps on the plane. The enumerating function equations of rooted biloopless maps, rooted nonseparable maps, rooted 2-edge-connected bipartite maps and rooted loopless Eulerian maps on the plane are derived by the chromatic sum function equation of rooted 2-edge-connected maps on the plane. Moreover, some explicit expressions of enumerating functions are also derived.

Chromatic sums of rooted triangulations on the projective plane

In this paper we study the chromatic sum functions for rooted nonseparable near-triangular maps on the projective plane. A chromatic sum equation for such maps is obtained.

Approximating Min Sum Set Cover

The input to the min sum set cover problem is a collection of n sets that jointly cover m elements. The output is a linear order on the sets, namely, in every time step from 1 to n exactly one set is chosen. For every element, this induces a first time step by which it is covered. The objective is to find a linear arrangement of the sets that minimizes the sum of these first time steps over all elements. We show that a greedy algorithm approximates min sum set cover within a ratio of 4. This result was implicit in work of Bar-Noy, Bellare, Halldorsson, Shachnai, and Tamir (1998) on chromatic sums, but we present a simpler proof. We also show that for every e > 0, achieving an approximation ratio of 4 – e is NP-hard. For the min sum vertex cover version of the problem (which comes up as a heuristic for speeding up solvers of semidefinite programs) we show that it can be approximated within a ratio of 2, and is NP-hard to approximate within some constant ρ > 1.

Chromatic sums of singular maps on some surfaces

A map is singular if each edge is on the same face on a surface (i.e., those have only one face on a surface). Because any map with loop is not colorable, all maps here are assumed to be loopless. In this paper provides the explicit expression of chromatic sum functions for rooted singular maps on the projective plane, the torus and the Klein bottle. From the explicit expression of chromatic sum functions of such maps, the explicit expression of enumerating functions of such maps are also derived.

Chromatic sums of nonseparable simple maps on the plane

In this paper we study the chromatic sum functions for rooted nonseparable simple maps on the plane. The chromatic sum function equation for such maps is obtained. The enumerating function equation of such maps is derived by the chromatic sum equation of such maps. From the chromatic sum equation of such maps, the enumerating function equation of rooted nonseparable simple bipartite maps on the plane is also derived.

On Chromatic Sums and Distributed Resource Allocation

This paper studies an optimization problem that arises in the context of distributed resource allocation: Given a conflict graph that represents the competition of processors over resources, we seek an allocation under which no two jobs with conflicting requirements are executed simultaneously. Our objective is to minimize the average response time of the system. In alternative formulation this is known as the Minimum Color Sum (MCS) problem (E. Kubicka and A. J. Schwenk, 1989. An introduction to chromatic sums, in “Proceedings of the ACM Computer Science Conference,” pp. 39–45.). We show that the algorithm based on finding iteratively a maximum independent set ( MaxIS ) is a 4-approximation to the MCS. This bound is tight to within a factor of 2. We give improved ratios for the classes of bipartite, bounded-degree, and line graphs. The bound generalizes to a 4 ρ -approximation of MCS for classes of graphs for which the maximum independent set problem can be approximated within a factor of ρ . On the other hand, we show that an n 1− ε -approximation is NP-hard, for some ε >0. For some instances of the resource allocation problem, such as the Dining Philosophers , an efficient solution requires edge coloring of the conflict graph. We introduce the Minimum Edge Color Sum (MECS) problem which is shown to be NP-hard. We show that a 2-approximation to MECS( G ) can be obtained distributively using compact coloring within O (log 2 n ) communication rounds

Chromatic sums revisited

This paper discusses some equations arising in the author's work on “chromatic sums”. The main results were presented in a series of papers extending from 1973 to 1982. The object here is to give a unified and simplified account of the elimination of unwanted variables from the initial equation, leading to the final identification of the desired chromatic sums as the coefficients in a power series satisfying a certain differential equation.

On chromatic and dichromatic sum equations

This paper provides firstly an exposition on the theory of chromatic sums for rooted planar triangulations founded by W.T. Tutte. Then, the generalization of the theory for rooted nonseparable planar maps is described. Further, a functional equation of the dichromatic sum functions for rooted nonseparable planar maps is also established by using the same decomposition lemmas as those in the chromatic case. However, it seems that the equations obtained here can not be directly derived from the dichromatic sum equation for rooted general planar maps which was discovered by W.T. Tutte as well.

On a pair of functional equations of combinatorial interest

The equations of the title appear in the author's paper “Chromatic Sums for Rooted Planar Triangulations, V: Special Equations.” (Canadian Journal of Mathematics, 26 (1974), 893–907). They appear in that paper as Equations (24) and (25). They are simultaneous equations for two unknown functionsl andy2 of two variablesy1 andz. A parameterμ is involved. The main result is that forμ = 2 cos (2π/n), wheren is a positive integer >1, the two equations can be reduced to a single equation (numbered (49)). Solutions of this are known forn <7. From such solutions we can expect to get information about the averaged chromatic polynomials of planar triangulations with a given number of triangles.

Chromatic Sums for Rooted Planar Triangulations, V: Special Equations

In I we obtained an equation, called the "chromatic equation", for the generating function g(x, y, z, λ). In II and III we obtained special equations, valid in the cases λ = τ + 1 and λ = 3 respectively, for the generating function l(y, z, λ), defined as the coefficient of x2 in g(x, y, z, λ). The argument was independent of that in I and no attempt was made to derive the new formulae from the chromatic equation.

Chromatic Sums for Rooted Planar Triangulations, IV: The Case λ = ∞

SummaryThe chromial P(M, λ) of a planar near-triangulation M has the leading term λv(M), where v(M) is the number of vertices of M. The problem of finding the number of rooted planar near-triangulations of a given class S, all supposed to have the same number of vertices, can be regarded as a special case of the problem of finding chromatic sums. We can sum P(M, λ) over the members of S, divide by the appropriate power of λ and let λ → ∞. We thus get the sum of the coefficient of the leading term of P(M, λ) for all M ∈ S, that is we get the number of members of S.

Chromatic Sums for Rooted Planar Triangulations, III: The Case λ = 3

SummaryIn this paper we are chiefly concerned with the chromatic sums we have called l and h, with colour-number 3. In this case h can be interpreted as enumerating the rooted Eulerian triangulations with a given number of faces, and l as enumerating such triangulations with a given number of faces and a given valency for the root-vertex. The series h has been determined already, by summation from the formula enumerating even slicings [3]. However our formula for l does not seem to have been published before, though it could presumably be derived in a similar way.

Chromatic Sums for Rooted Planar Triangulations II: The Case λ = τ + 1

SummaryIn an earlier paper [2] we denned the chromatic sums g, q, l and h. We determined the derivatives of these sums with respect to the colour-number λ at the special values λ = 1 and λ = 2. In the present paper we find parametric equations for h, l and q in the case λ = τ + 1, where τ is the golden ratio. We obtain h, l and the basic indeterminate z explicitly in terms of the parameter u, but for q we exhibit only a cubic equation with coefficients depending on u. We obtain an exact formula for the coefficients in h by applying Lagrange's theorem to the parametric equations.

Chromatic Sums for Rooted Planar Triangulations: The Cases λ = 1 and λ = 2

SummaryIn this paper we derive a functional equation whose solution would give the sum of the chromatic polynomialP(M,λ) over certain classes of rooted planar mapsMcalled “ triangulations” and “near-triangulations”. For an integral colour-number λ this sum is the number of λ-coloured rooted maps of the kind considered, but the sum can also be discussed for non-integral λ.