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

T-colorings, divisibility and the circular chromatic number

LetTbe a $T-set$, i.e., a finite set of nonnegative integers satisfying $0∈T$,and$Gbe$ a graph. In the paper we study relations between theT-edge spansesp $T(G)$ and $espd⊙T(G$), wheredis a positive integer and $d⊙T={0≤t≤d(maxT+ 1) :d|t⇒t/d∈T}$.We show that $espd⊙T(G)$ =$despT(G)−r$, wherer, $0≤r≤d−1$, is aninteger that depends onTandG. Next we focus on the $caseT={0}$ and show that $espd⊙{0}(G) =⌈d(χc(G)−1)⌉$,where $χc(G)$ is the circular chromatic number ofG. This result allows us toformulate several interesting conclusions that include a new formula for thecircular chromatic $numberχc(G)$ = $1 + inf{espd⊙{0}(G)/d:d≥1}$ $2R$ and a proof that the formula for the $T-edge$ span of powers of cycles, statedas conjecture in [Y. Zhao, W. He and R. Cao,The edge span ofT-coloringon graphCdn, Appl. Math. Lett. 19 (2006) 647–651], is true

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

Comprehending Gerasimov’s Perception of a Contemporary Conflict : The Way to Prevent Cyber Conflicts

Alongside with the appearance of the so far unknown reality called cyberspace, the new conditions of the course of conflicts emerged, consequently both the scientists as well as practitioners started to use the term cyber conflict. Unfortunately, presently there is no consistent, common view concerning a cyber conflict. The article presents a theoretical basis of cyber conflicts based on the research carried out by the authors. The article itself is an added value since it provides the suggestion and explanation of the perspective for the understanding of cyber conflicts through the prism of Gerasimov’s perception of a contemporary conflict. Moreover, it presents a new definition of a cyber conflict as the process being the system of activities. The authors also present the stages of a conflict according to Gerasimov, as well as the structure of a cyber conflict. Additionally, the article envisages the aspects of Russian attitude to conflict solving which are worth paying attention to. The presented article offers the perspective of the Russian understanding of the resolution of conflicts, it bridges the gap in research on cyber conflicts as well as assures a strong theoretical basis for the understanding of a Russian point of view on the solution of contemporary conflicts, which might be useful for counteracting cyber conflicts. The authors hold the view that the article is the incentive for further research on cyber conflicts during competition

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

Systemic character of the information process in electronic reconnaissance in the Land Forces of the Polish Armed Forces

The system of electronic reconnaissance of the Polish Land Forces produces data and reconnaissance information for end users depending on their needs. Producing a piece is carried out in the information process. Research conducted using the triangulation method showed that the information process in the Polish Land Forces’ electronic reconnaissance is a system of operation on reconnaissance data and information, not based on developed methodology with the application of specific information system resources. Therefore this system has elements connected by relations which are the subprocesses of collecting, processing, storing and sharing of reconnaissance data and information. This process is conditioned by primary and secondary factors. The efficiency of the information process in the Polish Armed Forces’ electronic reconnaissance is limited by technical, organizational and procedural barriers

Współdziałanie sił zbrojnych RP i policji dla zapewnienia cyberbezpieczeństwa infrastruktury krytycznej państwa w czasie działań hybrydowych prowadzonych na terenie RP

Nowoczesna infrastruktura krytyczna uzależniona jest od teleinformatyki, a jej bezpieczeństwo zależy od bezpieczeństwa w cyberprzestrzeni. W artykule autor omawia prawne i doktrynalne uwarunkowania współdziałania między Siłami Zbrojnymi Rzeczypospolitej Polskiej a polską Policją dla zapewnienia cyberbezpieczeństwa infrastruktury krytycznej państwa w czasie działań hybrydowych prowadzonych na terytorium Polski. Autor przedstawia także istotę cyberataków oraz charakterystykę takich działań w cyberprzestrzeni oraz z jej wykorzystaniem. W artykule omówione zostały również teoretyczne i praktyczne aspekty współdziałania Sił Zbrojnych Rzeczypospolitej Polskiej z polską Policją. Niniejszy artykuł jest zaproszeniem do naukowej debaty na przedstawiony temat