arXiv subjects

Janez Žerovnik

Publications and source records attributed to Janez Žerovnik.

12 recordsLinked to original sources

arXiv2026

$[k]$-Roman domination on cylindrical grids $C_m \Box P_n$

Roman domination and its higher-order extensions have attracted considerable attention due to their natural interpretation in terms of defensive resource allocation on networks. The recently introduced $[k]$-Roman domination framework unifies classical Roman, double, triple, and higher-strength protection schemes by allowing each fortified vertex to provide up to $k$ levels of support. In this paper, we investigate the $[k]$-Roman domination number $γ_{[k]R}(G)$ on cylindrical grids $C_m \Box P_n$. We relate $[k]$-Roman domination to efficient domination and establish exact values for regular graphs admitting an efficient dominating set; as a consequence, we obtain explicit values for broad families of toroidal grids and determine exactly when the cylindrical graphs $C_m\Box P_n$ admit an efficient dominating set. Building on these structural insights, we derive several upper bounds for $γ_{[k]R}(C_m \Box P_n)$ for small fixed values of $m$, accompanied by explicit labeling patterns that attain these bounds. All obtained bounds are systematically compared, revealing parameter ranges in which different constructions dominate depending on the value of $k$ and the length of the path. In addition, we present exact packing numbers for selected cylindrical graphs, which complement the domination results and enable further refinements via local weight reductions. Our results extend and unify known domination-type parameters on grid-like structures and highlight new regularities that emerge as the reinforcement strength increases.

math.CO↗

arXiv2026

Further results on $[k]$-Roman domination on cylindrical grids $C_m \Box P_n$

In this paper, we study the $[k]$-Roman domination number of cylindrical graphs $C_m \Box P_n$. Our analysis begins with a general lower bound based on local neighborhood constraints. We show that $γ_{[kR]}(C_m\Box P_n) > \frac{(k+1)mn}{5}.$ By exploiting the connection between $[k]$-Roman domination and efficient domination, we characterize the cylindrical graphs for which the extremal local configuration $f(N[x])=k+1$ for every $x\in V(C_m\Box P_n)$ can occur. We show that this happens precisely for so called efficient graphs. For fixed small values $m\in\{5,\ldots,8\}$, we construct explicit periodic $[k]$-Roman dominating functions that yield sparse upper bounds. These constructions are complemented by a general uniform upper bound and by packing-refined bounds. A systematic comparison of the resulting bounds shows how their relative strength depends on the parameter $k$ and on the length of the path.

math.CO↗

arXiv2026

Upper bounds for double Roman domination and $[k]$-Roman domination of cylindrical graphs $C_m \Box P_n$

Roman-type domination parameters form an important class of graph invariants that model protection and resource allocation problems on networks. Among them, $[k]$-Roman domination provides a unified framework that generalizes Roman, double Roman, and higher-order variants. In this paper we investigate the $[k]$-Roman domination number of cylindrical grids $C_m\Box P_n$ and derive several new constructive upper bounds. Our approach combines three complementary techniques: linear periodic constructions, uniform ceiling-type labelings, and packing-based refinements. We first analyze the case $C_9\Box P_n$, where these three families of bounds can be compared explicitly and their relative efficiency is shown to depend on the parameter $k$. We then extend the linear constructions to cylindrical grids whose circumference is a multiple of one of the values $3,\dots,9$, obtaining a unified family of upper bounds for $C_{rt}\Box P_n$. Motivated by the asymptotic behavior of these estimates, we further derive general upper bounds depending only on the residue class of $m$ modulo $5$, which apply to all cylindrical grids. As a consequence, we obtain explicit estimates for the double Roman domination number $γ_{[2]R}(C_m\Box P_n)$ and compare the resulting multiple-based constructions with the residue-class bounds. This comparison shows that the residue-class construction becomes asymptotically superior for all sufficiently large admissible circumferences, while several exceptional small cases remain better covered by tailored constructions.

math.CO↗

arXiv2024

The 2-rainbow domination number of Cartesian product of cycles

A $k$-rainbow dominating function ($k$RDF) of $G$ is a function that assigns subsets of $ \{1,2,...,k\}$ to the vertices of $G$ such that for vertices $v$ with $f(v)=\emptyset $ we have $\bigcup\nolimits_{u\in N(v)}f(u)=\{1,2,...,k\}$. The weight $w(f)$ of a $k$RDF $f$ is defined as $w(f)=\sum_{v\in V(G)}\left\vert f(v)\right\vert $. The minimum weight of a $k$RDF of $G$ is called the $k$-rainbow domination number of $G$, which is denoted by $γ_{rk}(G)$. In this paper, we study the 2-rainbow domination number of the Cartesian product of two cycles. Exact values are given for a number of infinite families and we prove lower and upper bounds for all other cases.

math.CO↗

arXiv2024

On rainbow domination of cubic graphs

The structure of minimal weight rainbow domination functions of cubic graphs are studied. Based on general observations for cubic graphs, generalized Petersen graphs $P(ck,k)$ are characterized whose 4- and 5-rainbow domination numbers equal the general lower bounds. As $t$-rainbow domination of cubic graphs for $t \ge 6$ is trivial, characterizations of such generalized Petersen graphs $P(ck,k)$ are known for all $t$-rainbow domination numbers.In addition, new upper bounds for 4- and 5-rainbow domination numbers that are valid for all $P(ck,k)$ are provided.

math.CO↗

arXiv2023

A fresh look to a randomized massively parallel graph coloring algorithm

Petford and Welsh introduced a sequential heuristic algorithm for (approximately) solving the NP-hard graph coloring problem. The algorithm is based on the antivoter model and mimics the behaviour of a physical process based on a multi-particle system of statistical mechanics. It was later shown that the algorithm can be implemented in massively parallel model of computation. The increase of processing power in recent years allow us to perform an extensive analysis of the algorithms on a larger scale, leading to possibility of a more comprehensive understanding of the behaviour of the algorithm including the phase transition phenomena.

math.CO↗

arXiv2020

Real Forms of the Complex Neumann System: Real Roots of Polynomial $U_{\cal S}(λ)$

The topology of Liouville sets of the real forms of the complex generic Neumann system depends indirectly on the roots of the special polynomial $U_{\cal S}(λ)$. For certain polynomials, the existence and positions of the real roots, according to the suitable parameters of the system, is not obvious. In the paper, a novel method for checking the existence and positions of the real roots of the polynomials $U_{\cal S}(λ)$ is given. The method and algorithm are based on searching of a positive solution of a system of linear equations. We provide a complete solution to the problem of existence of real roots for all special polynomials in case $n=2$. This is a step closer to determining the topology of the Liouville sets.

math-ph↗

arXiv2016

Improved approximative multicoloring of hexagonal graphs

In 1999, McDiarmid and Reed conjectued that the approximation ratio $9/8$ of multichromatic number to weighted clique number asymptotically is the best possible for general weighted hexagonal graphs. We prove that there is a proper multicoloring of $G$ that uses at most $15 \lfloor\frac{ω(G)}{12}\rfloor+ 18 $ colors improving the best previously known asimptotic ratio from 4/3 to 5/4.

math.CO↗

arXiv2016

Using Newton's method to model a spatial light distribution of a LED with attached secondary optics

In design of optical systems based on LED (Light emitting diode) technology, a crucial task is to handle the unstructured data describing properties of optical elements in standard formats. This leads to the problem of data fitting within an appropriate model. Newton's method is used as an upgrade of previously developed most promising discrete optimization heuristics showing improvement of both performance and quality of solutions. Experiment also indicates that a combination of an algorithm that finds promising initial solutions as a preprocessor to Newton's method may be a winning idea, at least on some datasets of instances.

cs.OH↗

arXiv2016

Self Similarities of the Tower of Hanoi Graphs and a proof of the Frame-Stewart Conjecture

Considering the symmetries and self similarity properties of the corresponding labeled graphs, it is shown that the minimal number of moves in the Tower of Hanoi game with $p =4$ pegs and $n \geq p$ disks satisfies the recursive formula $ F(p,n) = \min_{1\leq i \leq n-1} \{ 2F(p,i) + F(p-1,n-i) \} $ which proves the strong Frame-Stewart conjecture for the case $p=4$. The method can be generalized to $p>4$.

math.CO↗

arXiv2012

Improved upper bounds for vertex and edge fault diameters of Cartesian graph bundles

Mixed fault diameter of a graph $G$, $ \D_{(a,b)}(G)$, is the maximal diameter of $G$ after deletion of any $a$ vertices and any $b$ edges. Special cases are the (vertex) fault diameter $\D^V_{a} = \D_{(a,0)}$ and the edge fault diameter $\D^E_{a} = \D_{(0,a)}$. Let $G$ be a Cartesian graph bundle with fibre $F$ over the base graph $B$. We show that (1) $\D^V_{a+b+1}(G)\leq \D^V_{a}(F)+\D^V_{b}(B)$ when the graphs $F$ and $B$ are $k_F$-connected and $k_B$-connected, $0< a < k_F$, $0< b < k_B$, and provided that $\D_{(a-1,1)}(F)\leq \D^{V}_{a} (F)$ and $\D_{(b-1,1)}(B)\leq \D^{V}_{b} (B)$ and (2) $\D^E_{a+b+1}(G)\leq \D^E_{a}(F)+\D^E_{b}(B)$ when the graphs $F$ and $B$ are $k_F$-edge connected and $k_B$-edge connected, $0\leq a < k_F$, $0\leq b < k_B$, and provided that $\D^E_{a}(F)\geq 2$ and $\D^E_{b}(B)\geq 2$.

math.CO↗

arXiv2007

Best insertion algorithm for resource-constrained project scheduling problem

This paper considers heuristics for well known resource-constrained project scheduling problem (RCPSP). First a feasible schedule is constructed using randomized best insertion algorithm. The construction is followed by a local search where a new solution is generated as follows: first we randomly delete m activities from the list, which are then reinserted in the list in consecutive order. At the end of run, the schedule with the minimum makespan is selected. Experimental work shows very good results on standard test instances found in PSPLIB

cs.DM↗