Papers by Hanna Furmańczyk
arXiv (Cornell University), Jun 10, 2014
We study a new problem for cubic graphs: bipartization of a cubic graph Q by deleting sufficientl... more We study a new problem for cubic graphs: bipartization of a cubic graph Q by deleting sufficiently large independent set I. It can be expressed as follows: Given a connected n-vertex tripartite cubic graph Q = (V, E) with independence number α(Q), does Q contain an independent set I of size k such that Q − I is bipartite? We are interested for which value of k the answer to this question is affirmative. We prove constructively that if α(Q) ≥ 4n/10, then the answer is positive for each k fulfilling ⌊(n − α(Q))/2⌋ ≤ k ≤ α(Q). It remains an open question if a similar construction is possible for cubic graphs with α(Q) < 4n/10. Next, we show that this problem with α(Q) ≥ 4n/10 and k fulfilling inequalities ⌊n/3⌋ ≤ k ≤ α(Q) can be related to semi-equitable graph 3-coloring, where one color class is of size k, and the subgraph induced by the remaining vertices is equitably 2-colored. This means that Q has a coloring of type (k, ⌈(n − k)/2⌉, ⌊(n − k)/2⌋).
Information Processing Letters, May 1, 2008
In this note we observe that the problem of mixed graph coloring can be solved in linear time for... more In this note we observe that the problem of mixed graph coloring can be solved in linear time for trees, which improves the quadratic algorithm of Hansen et al. [
Ars Mathematica Contemporanea, Feb 5, 2016
A graph is equitably k-colorable if its vertices can be partitioned into k independent sets in su... more A graph is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest k for which such a coloring exists is known as the equitable chromatic number of G and it is denoted by χ = (G). In this paper the problem of determinig χ = for coronas of cubic graphs is studied. Although the problem of ordinary coloring of coronas of cubic graphs is solvable in polynomial time, the problem of equitable coloring becomes NP-hard for these graphs. We provide polynomially solvable cases of coronas of cubic graphs and prove the NP-hardness in a general case. As a by-product we obtain a simple linear time algorithm for equitable coloring of such graphs which uses χ = (G) or χ = (G) + 1 colors. Our algorithm is best possible, unless P=NP. Consequently, cubical coronas seem to be the only known class of graphs for which equitable coloring is harder than ordinary coloring.
Discrete Applied Mathematics, Mar 1, 2018
A k-coloring of a graph G = (V, E) is called semi-equitable if there exists a partition of its ve... more A k-coloring of a graph G = (V, E) is called semi-equitable if there exists a partition of its vertex set into independent subsets V 1 ,. .. , V k in such a way that |V 1 | / ∈ { |V |/k , |V |/k } and ||V i | − |V j || ≤ 1 for each i, j = 2,. .. , k. The color class V 1 is called non-equitable. In this note we consider the complexity of semi-equitable k-coloring, k ≥ 4, of the vertices of a cubic or subcubic graph G. In particular, we show that, given a n-vertex subcubic graph G and constants > 0, k ≥ 4, it is NP-complete to obtain a semi-equitable k-coloring of G whose non-equitable color class is of size s if s ≥ n/3 + n, and it is polynomially solvable if s ≤ n/3.
Szeregowanie zadań sprzężonych metodą kolorowania grafów
Automatyka / Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie, 2003
arXiv (Cornell University), Sep 27, 2020
An equitable coloring of a graph G is its (proper) vertex-coloring such that the sizes of any two... more An equitable coloring of a graph G is its (proper) vertex-coloring such that the sizes of any two color classes differ by at most one. In this paper, we consider the equitable coloring problem in block graphs. Recall that the latter are graphs in which each 2-connected component is a complete graph. The problem of equitable coloring remains hard in the class of block graphs. In this paper, we present some graph theoretic results relating various parameters. We use them in order to trace some algorithmic implications, mainly dealing with the fixed-parameter tractability of the problem.
Discrete Mathematics, Jun 1, 2009
We are interested in coloring the edges of a mixed graph, i.e., a graph containing unoriented and... more We are interested in coloring the edges of a mixed graph, i.e., a graph containing unoriented and oriented edges. This problem is related to a communication problem in job-shop scheduling systems. In this paper we give general bounds on the number of required colors and analyze the complexity status of this problem. In particular, we provide N Pcompleteness results for the case of outerplanar graphs, as well as for 3-regular bipartite graphs (even when only 3 colors are allowed, or when 5 colors are allowed and the graph is fully oriented). Special cases admitting polynomial-time solutions are also discussed.
Archives of Control Sciences, Mar 1, 2015
In the paper we consider the problems of equitable and semi-equitable coloring of vertices of cub... more In the paper we consider the problems of equitable and semi-equitable coloring of vertices of cubic graphs. We show that in contrast to the equitable coloring, which is easy, the problem of semi-equitable coloring is NP-complete within a broad spectrum of graph parameters. This affects the complexity of batch scheduling of unit-length jobs with cubic incompatibility graph on three uniform processors to minimize the makespan.
Discussiones Mathematicae Graph Theory, 2009
Springer eBooks, May 28, 2008
We consider the mixed graph coloring problem. A mixed graph GM = (V, E, A) is a graph with vertex... more We consider the mixed graph coloring problem. A mixed graph GM = (V, E, A) is a graph with vertex set V and containing edges (set E) and arcs (set A). An edge joining vertices i and j is denoted
arXiv (Cornell University), Apr 19, 2017
A graph G is equitably k-colorable if its vertices can be partitioned into k independent sets in ... more A graph G is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest integer k for which such a coloring exists is known as the equitable chromatic number of G and it is denoted by χ = (G). In this paper the problem of determinig the value of equitable chromatic number for multicoronas of cubic graphs G • l H is studied. The problem of ordinary coloring of multicoronas of cubic graphs is solvable in polynomial time. The complexity of equitable coloring problem is an open question for these graphs. We provide some polynomially solvable cases of cubical multicoronas and give simple linear time algorithms for equitable coloring of such graphs which use at most χ = (G • l H) + 1 colors in the remaining cases.
arXiv (Cornell University), Jun 27, 2021
In this paper the problem of scheduling of jobs on parallel machines under incompatibility relati... more In this paper the problem of scheduling of jobs on parallel machines under incompatibility relation is considered. In this model a binary relation between jobs is given and no two jobs that are in the relation can be scheduled on the same machine. In particular, we consider job scheduling under incompatibility relation forming bipartite graphs, under makespan optimality criterion, on uniform and unrelated machines. We show that no algorithm can achieve a good approximation ratio for uniform machines, even for a case of unit time jobs, under P ̸ = NP. We also provide an approximation algorithm that achieves the best possible approximation ratio, even for the case of jobs of arbitrary lengths pj, under the same assumption. Precisely, we present an O(n 1/2−ϵ) inapproximability bound, for any ϵ > 0; and pj-approximation algorithm, respectively. To enrich the analysis, bipartite graphs generated randomly according to Gilbert's model G n,n,p(n) are considered. For a broad class of p(n) functions we show that there exists an algorithm producing a schedule with makespan almost surely at most twice the optimum. Due to our knowledge, this is the first study of randomly generated graphs in the context of scheduling in the considered model. For unrelated machines, an FPTAS for R2|G = bipartite|Cmax is provided. We also show that there is no algorithm of approximation ratio O(n b p 1−ϵ max), even for Rm|G = bipartite|Cmax for m ≥ 3 and any ϵ > 0, b > 0, unless P = NP.
arXiv (Cornell University), Feb 14, 2015
In the paper we consider the problem of scheduling n identical jobs on 3 uniform machines with sp... more In the paper we consider the problem of scheduling n identical jobs on 3 uniform machines with speeds s 1 , s 2 , and s 3 to minimize the schedule length. We assume that jobs are subjected to some kind of mutual exclusion constraints, modeled by a cubic incompatibility graph. We show that if the graph is 2-chromatic then the problem can be solved in O(n 2) time. If the graph is 3-chromatic, the problem becomes NP-hard even if s 1 > s 2 = s 3. However, in this case there exists a 4/3-approximation algorithm running in O(n 3) time. Moreover, this algorithm solves the problem almost surely to optimality if 3s 1 /4 ≤ s 2 = s 3 .
The Complexity of Equitable Vertex Coloring of Graphs
Journal of Applied Computer Science, 2005
Bulletin of The Polish Academy of Sciences-technical Sciences, Feb 1, 2017
In the paper we consider the problem of scheduling n identical jobs on 4 uniform machines with sp... more In the paper we consider the problem of scheduling n identical jobs on 4 uniform machines with speeds s 1 ¸ s 2 ¸ s 3 ¸ s 4 , respectively. Our aim is to find a schedule with a minimum possible length. We assume that jobs are subject to some kind of mutual exclusion constraints modeled by a bipartite incompatibility graph of degree ∆, where two incompatible jobs cannot be processed on the same machine. We show that the general problem is NP-hard even if s 1 = s 2 = s 3. If, however, ∆ • 4 and s 1 ¸ 12s 2 , s 2 = s 3 = s 4 , then the problem can be solved to optimality in time O(n 1.5). The same algorithm returns a solution of value at most 2 times optimal provided that s 1 ¸ 2s 2. Finally, we study the case s 1 ¸ s 2 ¸ s 3 = s 4 and give a 32/15-approximation algorithm running also in O(n 1.5) time.
Discrete Applied Mathematics, 2018
In the paper we consider the problem of scheduling n identical jobs on 3 uniform machines with sp... more In the paper we consider the problem of scheduling n identical jobs on 3 uniform machines with speeds s 1 , s 2 , and s 3 to minimize the schedule length. We assume that jobs are subjected to some kind of mutual exclusion constraints, modeled by a cubic incompatibility graph. We show that if the graph is 2-chromatic then the problem can be solved in O(n 2) time. If the graph is 3-chromatic, the problem becomes NP-hard even if s 1 > s 2 = s 3. However, in this case there exists a 10/7-approximation algorithm running in O(n 3) time. Moreover, this algorithm solves the problem almost surely to optimality if 3s 1 /4 ≤ s 2 = s 3 .
arXiv (Cornell University), Aug 23, 2022
An adjacent vertex distinguishing total k-coloring f of a graph G is a proper total k-coloring of... more An adjacent vertex distinguishing total k-coloring f of a graph G is a proper total k-coloring of G such that no pair of adjacent vertices has the same color sets, where the color set at a vertex v, C G f (v), is {f (v)} ∪ {f (vu)|u ∈ V (G), vu ∈ E(G)}. In 2005 Zhang et al. posted the conjecture (AVDTCC) that every simple graph G has adjacent vertex distinguishing total (∆(G) + 3)-coloring. In this paper we confirm the conjecture for many coronas, in particular for generalized, simple and l-coronas of graphs, not relating the results to particular graph classes.
arXiv (Cornell University), Jul 12, 2022
The problem of scheduling jobs on parallel machines (identical, uniform, or unrelated), under inc... more The problem of scheduling jobs on parallel machines (identical, uniform, or unrelated), under incompatibility relation modeled as a block graph, under the makespan optimality criterion, is considered in this paper. No two jobs that are in the relation (equivalently in the same block) may be scheduled on the same machine in this model. The presented model stems from a well-established line of research combining scheduling theory with methods relevant to graph coloring. Recently, cluster graphs and their extensions like block graphs were given additional attention. We complement hardness results provided by other researchers for block graphs by providing approximation algorithms. In particular, we provide a 2-approximation algorithm for P |G = block graph|C max and a PTAS for the case when the jobs are unit time in addition. In the case of uniform machines, we analyze two cases. The first one is when the number of blocks is bounded, i.e. Q|G = k-block graph|C max. For this case, we provide a PTAS, improving upon results presented by D. Page and R. Solis-Oba. The improvement is twofold: we allow richer graph structure, and we allow the number of machine speeds to be part of the input. Due to strong NP-hardness of Q|G = 2-clique graph|C max , the result establishes the approximation status of Q|G = k-block graph|C max. The PTAS might be of independent interest because the problem is tightly related to the NUMERICAL k-DIMENSIONAL MATCHING WITH TARGET SUMS problem. The second case that we analyze is when the number of blocks is arbitrary, but the number of cut-vertices is bounded and jobs are of unit time. In this case, we present an exact algorithm. In addition, we present an FPTAS for graphs with bounded treewidth and a bounded number of unrelated machines.
Discussiones Mathematicae Graph Theory, 2017
A graph is equitably k-colorable if its vertices can be partitioned into k independent sets in su... more A graph is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the numbers of vertices in any two sets differ by at most one. The smallest k for which such a coloring exists is known as the equitable chromatic number of G and denoted by χ = (G). It is known that the problem of computation of χ = (G) is NP-hard in general and remains so for corona graphs. In this paper we consider the same model of coloring in the case of corona multiproducts of graphs. In particular, we obtain some results regarding the equitable chromatic number for the l-corona product G • l H, where G is an equitably 3-or 4-colorable graph and H is an r-partite graph, a cycle or a complete graph. Our proofs are mostly constructive in that they lead to polynomial algorithms for equitable coloring of such graph products provided that there is given an equitable coloring of G. Moreover, we confirm the Equitable Coloring Conjecture for 1 Research has been partially supported by Narodowe Centrum Nauki under contract DEC-2011/02/A/ST6/00201
Discussiones Mathematicae Graph Theory, 2021
The minimum number of total independent partition sets of V ∪ E of a graph G = (V, E) is called t... more The minimum number of total independent partition sets of V ∪ E of a graph G = (V, E) is called the total chromatic number of G, denoted by χ ′′ (G). If the difference between cardinalities of any two total independent sets is at most one, then the minimum number of total independent partition sets of V ∪ E is called the equitable total chromatic number, and is denoted by χ ′′ = (G). In this paper we consider equitable total coloring of coronas of cubic graphs, G•H. It turns out that independently on the values of equitable total chromatic number of factors G and H, equitable total chromatic number of corona G • H is equal to ∆(G • H) + 1. Thereby, we confirm Total Coloring Conjecture (TCC), posed by Behzad in 1964, and Equitable Total Coloring Conjecture (ETCC), posed by Wang in 2002, for coronas of cubic graphs. As a direct consequence we get that all coronas of cubic graphs are of Type 1.
Uploads
Papers by Hanna Furmańczyk