The complexity of the T-coloring problem for graphs with small degree

AbstractIn the paper we consider a generalized vertex coloring model, namely T-coloring. For a given finite set T of nonnegative integers including 0, a proper vertex coloring is called a T-coloring if the distance of the colors of adjacent vertices is not an element of T. This problem is a generalization of the classic vertex coloring and appeared as a model of the frequency assignment problem. We present new results concerning the complexity of T-coloring with the smallest span on graphs with small degree Δ. We distinguish between the cases that appear to be polynomial or NP-complete. More specifically, we show that our problem is polynomial on graphs with Δ⩽2 and in the case of k-regular graphs it becomes NP-hard even for every fixed T and every k>3. Also, the case of graphs with Δ=3 is under consideration. Our results are based on the complexity properties of the homomorphism of graphs

Algorithms for testing security in graphs

W niniejszym artykule przedstawiamy metodę weryfikowania bezpieczeństwa zbioru w grafie, dającą wysokie prawdopodobieństwo poprawnej weryfikacji. Problemem jest określenie, czy dla danego grafu G oraz podzbioru S zbioru wierzchołków tego grafu zbiór S jest bezpieczny, to znaczy każdy jego podzbiór X spełnia warunek: |N[X] ∩ S| ≥ |N[X] \ S|, gdzie N[X] jest domkniętym sąsiedztwem zbioru X w grafie G. Zaprojektowaliśmy pseudotester o wielomianowej złożoności obliczeniowej dla decyzyjnego problemubezpieczeństwa zbioru w grafie wykorzystując m.in. koncepcję symulowanego wyżarzania. Wykonaliśmy testy dla grafów, w których podgraf indukowany przez zbiór S jest drzewem lub grafem ograniczonego stopnia (przez 3 oraz 4). Z uwagi na coNP-zupełność problemu bezpieczeństwa zaproponowane przez nas podejście jest uogólnieniem koncepcji testowania własności znanej z literatury. In this paper we propose new algorithmic methods giving with a high probability the correct answer to the decision problem of security in graphs. For a given graph G and a subset S of a vertex set of G we have to decide whether S is secure, i.e. every subset X of S fulfils the condition: |N[X] S| |N[X] \ S|, where N[X] is a closed neighbourhood of X in graph G. We constructed a polynomial time property pseudotester based on the heuristic using simulated annealing and tested it on graphs with induced small subgraphs G[S] being trees or graphs with a bounded degree (by 3 or 4). Our approach is a generalization of the concept of property testers known from the subjectliterature, but we applied our concepts to the coNP-complete problem

The complexity of the L(p,q)-labeling problem for bipartite planar graphs of small degree

AbstractGiven a simple graph G, by an L(p,q)-labeling of G we mean a function c that assigns nonnegative integers to its vertices in such a way that if two vertices u, v are adjacent then |c(u)−c(v)|≥p, and if they are at distance 2 then |c(u)−c(v)|≥q. The L(p,q)-labeling problem can be defined as follows: given a graph G and integer t, determine whether there exists an L(p,q)-labeling c of G such that c(V)⊆{0,1,…,t}. In the paper we show that the problem is NP-complete even when restricted to bipartite planar graphs of small maximum degree and for relatively small values of t. More precisely, we prove that: (1)if p<3q then the problem is NP-complete for bipartite planar graphs of maximum degree Δ≤3 and t=p+max{2q,p};(2)if p=3q then the problem is NP-complete for bipartite planar graphs of maximum degree Δ≤4 and t=6q;(3)if p>3q then the problem is NP-complete for bipartite planar graphs of maximum degree Δ≤4 and t=p+5q.In particular, these results imply that the L(2,1)-labeling problem in planar graphs is NP-complete for t=4, and that the L(p,q)-labeling problem in graphs of maximum degree Δ≤4 is NP-complete for all values of p and q, thus answering two well-known open questions

On Efficient Coloring of Chordless Graphs

We are given a simple graph G = (V, E). Any edge e ∈ E is a chord in a path P ⊆ G (cycle C ⊆ G) iff a graph obtained by joining e to path P (cycle C) has exactly two vertices of degree 3. A class of graphs without any chord in paths (cycles) we call path-chordless (cycle-chordless). We will prove that recognizing and coloring of these graphs can be done in O(n2) and O(n) time, respectively. Our study was motivated by a wide range of applications of the graph coloring problem in coding theory, time tabling and scheduling, frequency assignment, register allocation and many other areas

Interval Incidence Coloring of Subcubic Graphs

In this paper we study the problem of interval incidence coloring of subcubic graphs. In [14] the authors proved that the interval incidence 4-coloring problem is polynomially solvable and the interval incidence 5-coloring problem is NP-complete, and they asked if Xii(G) ≤ 2Δ(G) holds for an arbitrary graph G. In this paper, we prove that an interval incidence 6-coloring always exists for any subcubic graph G with Δ(G) = 3

On Incidence Coloring of Complete Multipartite and Semicubic Bipartite Graphs

In the paper, we show that the incidence chromatic number χi of a complete k-partite graph is at most Δ + 2 (i.e., proving the incidence coloring conjecture for these graphs) and it is equal to Δ + 1 if and only if the smallest part has only one vertex (i.e., Δ = n − 1). Formally, for a complete k-partite graph G = Kr1,r2,...,rk with the size of the smallest part equal to r1 ≥ 1 we have χi(G)={Δ(G)+1if r1=1,Δ(G)+2if r1>1.\chi _i (G) = \left\{ {\matrix{{\Delta (G) + 1} & {{\rm{if}}\;r_1 = 1,} \cr {\Delta (G) + 2} & {{\rm{if}}\;r_1 > 1.} \cr } } \right. In the paper we prove that the incidence 4-coloring problem for semicubic bipartite graphs is -complete, thus we prove also the -completeness of L(1, 1)-labeling problem for semicubic bipartite graphs. Moreover, we observe that the incidence 4-coloring problem is -complete for cubic graphs, which was proved in the paper [12] (in terms of generalized dominating sets)

A polynomial algorithm for finding T-span of generalized cacti

AbstractIt has been known for years that the problem of computing the T-span is NP-hard in general. Recently, Giaro et al. (Discrete Appl. Math., to appear) showed that the problem remains NP-hard even for graphs of degree Δ⩽3 and it is polynomially solvable for graphs with degree Δ⩽2. Herein, we extend the latter result. We introduce a new class of graphs which is large enough to contain paths, cycles, trees, cacti, polygon trees and connected outerplanar graphs. Next, we study the properties of graphs from this class and prove that the problem of computing the T-span for these graphs is polynomially solvable