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Riste Škrekovski

Publications and source records attributed to Riste Škrekovski.

At least 37 records · Page 2Linked to original sources

arXiv2025

Invitation to the subpath number

In this paper we count all the subpaths of a given graph G; including the subpaths of length zero, and we call this quantity the subpath number of G. The subpath number is related to the extensively studied number of subtrees, as it can be considered as counting subtrees with the additional requirement of maximum degree being two. We first give the explicit formula for the subpath number of trees and unicyclic graphs. We show that among connected graphs on the same number of vertices, the minimum of the subpath number is attained for any tree and the maximum for the complete graph. Further, we show that the complete bipartite graph with partite sets of almost equal size maximizes the subpath number among all bipartite graphs. The explicit formula for cycle chains, i.e. graphs in which two consecutive cycles share a single edge, is also given. This family of graphs includes the unbranched catacondensed benzenoids which implies a possible application of the result in chemistry. The paper is concluded with several directions for possible further research where several conjectures are provided.

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arXiv2024

Some results on $σ_{t}$-irregularity

The $σ_{t}$-irregularity (or sigma total index) is a graph invariant which is defined as $σ_{t}(G)=\sum_{\{u,v\}\subseteq V(G)}(d(u)-d(v))^{2},$ where $d(z)$ denotes the degree of $z$. This irregularity measure was proposed by R\' {e}ti [Appl. Math. Comput. 344-345 (2019) 107-115], and recently rediscovered by Dimitrov and Stevanović [Appl. Math. Comput. 441 (2023) 127709]. In this paper we remark that $σ_{t}(G)=n^{2}\cdot Var(G)$, where $Var(G)$ is the degree variance of the graph. Based on this observation, we characterize irregular graphs with maximum $σ_{t}$-irregularity. We show that among all connected graphs on $n$ vertices, the split graphs $S_{\lceil\frac{n}{4}\rceil, \lfloor\frac{3n}{4}\rfloor }$ and $S_{\lfloor\frac{n}{4}\rfloor, \lceil\frac{3n}{4}\rceil }$ have the maximum $σ_{t}$-irregularity, and among all complete bipartite graphs on $n$ vertices, either the complete bipartite graph $K_{\lfloor\frac{n}{4}(2-\sqrt{2})\rfloor, \lceil\frac{n}{4}(2+\sqrt{2})\rceil }$ or $K_{\lceil\frac{n}{4}(2-\sqrt{2})\rceil, \lfloor\frac{n}{4}(2+\sqrt{2})\rfloor }$ has the maximum sigma total index. Moreover, various upper and lower bounds for $σ_{t}$-irregularity are provided; in this direction we give a relation between the graph energy $\mathcal{E}(G)$ and sigma total index $σ_{t}(G)$ and give another proof of two results by Dimitrov and Stevanović. Applying Fiedler's characterization of the largest and the second smallest Laplacian eigenvalue of the graph, we also establish new relationships between $σ_{t}$ and $σ$. We conclude the paper with two conjectures.

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arXiv2024

Extremizing antiregular graphs by modifying total $σ$-irregularity

The total $σ$-irregularity is given by $ σ_t(G) = \sum_{\{u,v\} \subseteq V(G)} \left(d_G(u) - d_G(v)\right)^2, $ where $d_G(z)$ indicates the degree of a vertex $z$ within the graph $G$. It is known that the graphs maximizing $σ_{t}$-irregularity are split graphs with only a few distinct degrees. Since one might typically expect that graphs with as many distinct degrees as possible achieve maximum irregularity measures, we modify this invariant to $ \IR(G)= \sum_{\{u,v\} \subseteq V(G)} |d_G(u)-d_G(v)|^{f(n)}, $ where $n=|V(G)|$ and $f(n)>0$. We study under what conditions the above modification obtains its maximum for antiregular graphs. We consider general graphs, trees, and chemical graphs, and accompany our results with a few problems and conjectures.

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arXiv2024

Fault tolerance of metric basis can be expensive

A set of vertices S is a resolving set of a graph G; if for every pair of vertices x and y in G, there exists a vertex s in S such that x and y differ in distance to s. A smallest resolving set of G is called a metric basis. The metric dimension dim(G) is the cardinality of a metric basis of G. The notion of a metric basis is applied to the problem of placing sensors in a network, where the problem of sensor faults can arise. The fault-tolerant metric dimension ftdim(G) is the cardinality of a smallest resolving set S such that S\{s} remains a resolving set of G for every s in S. A natural question is how much more sensors need to be used to achieve a fault-tolerant metric basis. It is known in literature that there exists an upper bound on ftdim(G) which is exponential in terms of dim(G); i.e. ftdim(G) <= dim(G)(1+2^(5dim(G)-1)). In this paper, we construct graphs G with ftdim(G) = dim(G)+2^(dim(G)-1) for any value of dim(G), so the exponential upper bound is necessary. We also extend these results to the k-metric dimension which is a generalization of the fault-tolerant metric dimension. First, we establish a similar exponential upper bound on dim(k+1)(G) in terms of dim(k)(G); and then we show that there exists a graph for which dim(k+1)(G) is indeed exponential. For a possible further work, we leave the gap between the bounds to be reduced.

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arXiv2024

On strong odd colorings of graphs

A strong odd coloring of a simple graph $G$ is a proper coloring of the vertices of $G$ such that for every vertex $v$ and every color $c$, either $c$ is used an odd number of times in the open neighborhood $N_G(v)$ or no neighbor of $v$ is colored by $c$. The smallest integer $k$ for which $G$ admits a strong odd coloring with $k$ colors is the strong odd chromatic number, $χ_{soc}(G)$. These coloring notion and graph parameter were recently defined in [H. Kwon and B. Park, Strong odd coloring of sparse graphs, ArXiv:2401.11653v2]. We answer a question raised by the originators concerning the existence of a constant bound for the strong odd chromatic number of all planar graphs. We also consider strong odd colorings of trees, unicyclic graphs and graph products.

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arXiv2024

Normal 5-edge-coloring of some snarks superpositioned by Flower snarks

An edge e is normal in a proper edge-coloring of a cubic graph G if the number of distinct colors on four edges incident to e is 2 or 4: A normal edge-coloring of G is a proper edge-coloring in which every edge of G is normal. The Petersen Coloring Conjecture is equivalent to stating that every bridgeless cubic graph has a normal 5-edge-coloring. Since every 3-edge-coloring of a cubic graph is trivially normal, it is suficient to consider only snarks to establish the conjecture. In this paper, we consider a class of superpositioned snarks obtained by choosing a cycle C in a snark G and superpositioning vertices of C by one of two simple supervertices and edges of C by superedges Hx;y, where H is any snark and x; y any pair of nonadjacent vertices of H: For such superpositioned snarks, two suficient conditions are given for the existence of a normal 5-edge-coloring. The first condition yields a normal 5-edge-coloring for all hypohamiltonian snarks used as superedges, but only for some of the possible ways of connecting them. In particular, since the Flower snarks are hypohamiltonian, this consequently yields a normal 5-edge-coloring for many snarks superpositioned by the Flower snarks. The second sufficient condition is more demanding, but its application yields a normal 5-edge-colorings for all superpositions by the Flower snarks. The same class of snarks is considered in [S. Liu, R.-X. Hao, C.-Q. Zhang, Berge{Fulkerson coloring for some families of superposition snarks, Eur. J. Comb. 96 (2021) 103344] for the Berge-Fulkerson conjecture. Since we established that this class has a Petersen coloring, this immediately yields the result of the above mentioned paper.

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arXiv2024

On regular graphs with Šoltés vertices

Let $W(G)$ be the Wiener index of a graph $G$. We say that a vertex $v \in V(G)$ is a Šoltés vertex in $G$ if $W(G - v) = W(G)$, i.e. the Wiener index does not change if the vertex $v$ is removed. In 1991, Šoltés posed the problem of identifying all connected graphs $G$ with the property that all vertices of $G$ are Šoltés vertices. The only such graph known to this day is $C_{11}$. As the original problem appears to be too challenging, several relaxations were studied: one may look for graphs with at least $k$ Šoltés vertices; or one may look for $α$-Šoltés graphs, i.e. graphs where the ratio between the number of Šoltés vertices and the order of the graph is at least $α$. Note that the original problem is, in fact, to find all $1$-Šoltés graphs. We intuitively believe that every $1$-Šoltés graph has to be regular and has to possess a high degree of symmetry. Therefore, we are interested in regular graphs that contain one or more Šoltés vertices. In this paper, we present several partial results. For every $r\ge 1$ we describe a construction of an infinite family of cubic $2$-connected graphs with at least $2^r$ Šoltés vertices. Moreover, we report that a computer search on publicly available collections of vertex-transitive graphs did not reveal any $1$-Šoltés graph. We are only able to provide examples of large $\frac{1}{3}$-Šoltés graphs that are obtained by truncating certain cubic vertex-transitive graphs. This leads us to believe that no $1$-Šoltés graph other than $C_{11}$ exists.

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arXiv2024

Criticality in Sperner's Lemma

We answer a question posed by T. Gallai in 1969 concerning criticality in Sperner's lemma, listed as Problem 9.14 in the collection of Jensen and Toft [Graph coloring problems, John Wiley & Sons, Inc., New York, 1995]. Sperner's lemma states that if a labelling of the vertices of a triangulation of the $d$-simplex $Δ^d$ with labels $1, 2, \ldots, d+1$ has the property that (i) each vertex of $Δ^d$ receives a distinct label, and (ii) any vertex lying in a face of $Δ^d$ has the same label as one of the vertices of that face, then there exists a rainbow facet (a facet whose vertices have pairwise distinct labels). For $d\leq 2$, it is not difficult to show that for every facet $σ$, there exists a labelling with the above properties where $σ$ is the unique rainbow facet. For every $d\geq 3$, however, we construct an infinite family of examples where this is not the case, which implies the answer to Gallai's question as a corollary. The construction is based on the properties of a $4$-polytope which had been used earlier to disprove a claim of T. S. Motzkin on neighbourly polytopes.

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arXiv2024

Domination number of modular product graphs

The modular product $G\diamond H$ of graphs $G$ and $H$ is a graph on vertex set $V(G)\times V(H)$. Two vertices $(g,h)$ and $(g^{\prime},h^{\prime})$ of $G\diamond H$ are adjacent if $g=g^{\prime}$ and $hh^{\prime}\in E(H)$, or $gg^{\prime}\in E(G)$ and $h=h^{\prime}$, or $gg^{\prime}\in E(G)$ and $hh^{\prime}\in E(H)$, or (for $g\neq g^{\prime}$ and $h\neq h^{\prime}$) $gg^{\prime}\notin E(G)$ and $hh^{\prime}\notin E(H)$. A set $D\subseteq V(G)$ is a dominating set of $G$ if every vertex outside of $D$ contains a neighbor in $D$. A set $D\subseteq V(G)$ is a total dominating set of $G$ if every vertex of $G$ contains a neighbor in $D$. The domination number $γ(G)$ (resp. total domination number $γ_{t}(G)$) of $G$ is the minimum cardinality of a dominating set (resp. total dominating set) of $G$. In this work we give several upper and lower bounds for $γ(G\diamond H)$ in terms of $γ(G),$ $γ(H)$, $γ_{t}(\overline{G})$ and $γ_{t}(\overline{H})$, where $\overline{G}$ is the complement graph of $G$. Further, we fully describe graphs where $γ(G\diamond H)=k$ for $k\in\{1,2,3\}$. Several conditions on $G$ and $H$ under which $γ(G\diamond H)$ is at most $4$ and $5$ are also given. A new type of simultaneous domination $\barγ(G)$, defined as the smallest number of vertices that dominates $G$ and totally dominates the complement of $G,$ emerged as useful and we believe it could be of independent interest. We conclude the paper by proposing few directions for possible further research.

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arXiv2024

Proper Z4 x Z2-colorings: structural characterization with application to some snarks

A proper abelian coloring of a cubic graph G by a finite abelian group A is any proper edge-coloring of G by the non-zero elements of A such that the sum of the colors of the three edges incident to any vertex v of G equals zero. It is known that cyclic groups of order smaller than 10 do not color all bridgeless cubic graphs, and that all abelian groups of order at least 12 do. This leaves the question open for the four so called exceptional groups Z4 x Z2, Z3 x Z3, Z10 and Z11 for snarks. It is conjectured in literature that every cubic graph has a proper abelian coloring by each exceptional group and it is further known that the existence of a proper Z4 x Z2-coloring of G implies the existence of a proper coloring of G by all the remaining exceptional groups. In this paper, we give a characterization of a proper Z4 x Z2-coloring in terms of the existence of a matching M in a 2-factor F of G with particular properties. Moreover, in order to modify an arbitrary matching M so that it meets the requirements of the characterization, we first introduce an incidence structure of the cycles of F in relation to the cycles of G - M. Further, we provide a sufficient condition under which M can be modified into a desired matching in terms of particular properties of the introduced incidence structure. We conclude the paper by applying the results to some oddness two snarks, in particular to permutation snarks. We believe that the approach of this paper with some additional refinements extends to larger classes of snarks, if not to all in general.

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arXiv2023

Normal 5-edge coloring of some more snarks superpositioned by the Petersen graph

A normal 5-edge-coloring of a cubic graph is a coloring such that for every edge the number of distinct colors incident to its end-vertices is 3 or 5 (and not 4). The well known Petersen Coloring Conjecture is equivalent to the statement that every bridgeless cubic graph has a normal 5-edge-coloring. All 3-edge-colorings of a cubic graph are obviously normal, so in order to establish the conjecture it is sufficient to consider only snarks. In our previous paper [J. Sedlar, R. Škrekovski, Normal 5-edge-coloring of some snarks superpositioned by the Petersen graph, Applied Mathematics and Computation 467 (2024) 128493], we considered superpositions of any snark G along a cycle C by two simple supervertices and by the superedge obtained from the Petersen graph, but only for some of the possible ways of connecting supervertices and superedges. The present paper is a continuation of that paper, herein we consider superpositions by the Petersen graph for all the remaining connections and establish that for all of them the Petersen Coloring Conjecture holds.

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arXiv2023

Resolving vertices of graphs with differences

The classical (vertex) metric dimension of a graph G is defined as the cardinality of a smallest set S in V (G) such that any two vertices x and y from G have different distances to least one vertex from S: The k-metric dimension is a generalization of that notion where it is required that any pair of vertices has different distances to at least k vertices from S: In this paper, we introduce the weak k-metric dimension of a graph G; which is defined as the cardinality of a smallest set of vertices S such that the sum of the distance differences from any pair of vertices to all vertices of S is at least k: This dimension is "stronger" than the classical metric dimension, yet "weaker" than k-metric dimension, and it can be formulated as an ILP problem. The maximum k for which the weak k-metric dimension is defined is denoted by kappa(G). We first prove several properties of the weak k-metric dimension regarding the presence of true or false twin vertices in a graph. Using those properties, the kappa(G) is found for some basic graph classes, such as paths, stars, cycles, and complete (bipartite) graphs. We also find kappa(G) for trees and grid graphs using the observation that the distance difference increases by the increase of the cardinality of a set S. For all these graph classes we further establish the exact value of the weak k-metric dimension for all k <= kappa(G).

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arXiv2023

Normal 5-edge-coloring of some snarks superpositioned by the Petersen graph

In a (proper) edge-coloring of a bridgeless cubic graph G an edge e is rich (resp. poor) if the number of colors of all edges incident to end-vertices of e is 5 (resp. 3). An edge-coloring of G is is normal if every edge of G is either rich or poor. In this paper we consider snarks ~G obtained by a simple superposition of edges and vertices of a cycle C in a snark G: For an even cycle C we show that a normal coloring of G can be extended to a normal coloring of ~G without changing colors of edges outside C in G: An interesting remark is that this is in general impossible for odd cycles, since the normal coloring of a Petersen graph P10 cannot be extended to a superposition of P10 on a 5-cycle without changing colors outside the 5-cycle. On the other hand, as our colorings of the superpositioned snarks introduce 18 or more poor edges, we are inclined to believe that every bridgeless cubic graph istinct from P10 has a normal coloring with at least one poor edge and possibly with at least 6 if we also exclude the Petersen graph with one vertex truncated.

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arXiv2023

Selected topics on Wiener index

The Wiener index is defined as the sum of distances between all unordered pairs of vertices in a graph. It is one of the most recognized and well-researched topological indices, which is on the other hand still a very active area of research. This work presents a natural continuation of the paper Mathematical aspects of Wiener index (Ars Math. Contemp., 2016) in which several interesting open questions on the topic were outlined. Here we collect answers gathered so far, give further insights on the topic of extremal values of Wiener index in different settings, and present further intriguing problems and conjectures.

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arXiv2023

Odd edge-colorings of subdivisions of odd graphs

An odd graph is a finite graph all of whose vertices have odd degrees. Given graph $G$ is decomposable into $k$ odd subgraphs if its edge set can be partitioned into $k$ subsets each of which induces an odd subgraph of $G$. The minimum value of $k$ for which such a decomposition of $G$ exists is the odd chromatic index, $χ_{o}'(G)$, introduced by Pyber (1991). For every $k\geqχ_{o}'(G)$, the graph $G$ is said to be odd $k$-edge-colorable. Apart from two particular exceptions, which are respectively odd $5$- and odd $6$-edge-colorable, the rest of connected loopless graphs are odd $4$-edge-colorable, and moreover one of the color classes can be reduced to size $\leq2$. In addition, it has been conjectured that an odd $4$-edge-coloring with a color class of size at most $1$ is always achievable. Atanasov et al. (2016) characterized the class of subcubic graphs in terms of the value $χ_{o}'(G)\leq4$. In this paper, we extend their result to a characterization of all subdivisions of odd graphs in terms of the value of the odd chromatic index. This larger class $\mathcal{S}$ is of a particular interest as it collects all `least instances' of non-odd graphs. As a prelude to our main result, we show that every connected graph $G\in \mathcal{S}$ requiring the maximum number of four colors, becomes odd $3$-edge-colorable after removing a certain edge. Thus, we provide support for the mentioned conjecture by proving it for all subdivisions of odd graphs. The paper concludes with few problems for possible further work.

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arXiv2022

Remarks on odd colorings of graphs

A proper vertex coloring $φ$ of graph $G$ is said to be odd if for each non-isolated vertex $x\in V(G)$ there exists a color $c$ such that $φ^{-1}(c)\cap N(x)$ is odd-sized. The minimum number of colors in any odd coloring of $G$, denoted $χ_o(G)$, is the odd chromatic number. Odd colorings were recently introduced in [M.~Petruševski, R.~Škrekovski: \textit{Colorings with neighborhood parity condition}]. Here we discuss various basic properties of this new graph parameter, characterize acyclic graphs and hypercubes in terms of odd chromatic number, establish several upper bounds in regard to degenericity or maximum degree, and pose several questions and problems.

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arXiv2022

Local Irregularity Conjecture vs. cacti

A graph is locally irregular if the degrees of the end-vertices of every edge are distinct. An edge coloring of a graph G is locally irregular if every color induces a locally irregular subgraph of G. A colorable graph G is any graph which admits a locally irregular edge coloring. The locally irregular chromatic index X'irr(G) of a colorable graph G is the smallest number of colors required by a locally irregular edge coloring of G. The Local Irregularity Conjecture claims that all colorable graphs require at most 3 colors for locally irregular edge coloring. Recently, it has been observed that the conjecture does not hold for the bow-tie graph B [7]. Cacti are important class of graphs for this conjecture since B and all non-colorable graphs are cacti. In this paper we show that for every colorable cactus graph G != B it holds that X'irr(G) <= 3. This makes us to believe that B is the only colorable graph with X'irr(B) > 3, and consequently that B is the only counterexample to the Local Irregularity Conjecture.

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arXiv2022

A note on the locally irregular edge colorings of cacti

A graph is locally irregular if the degrees of the end-vertices of every edge are distinct. An edge coloring of a graph G is locally irregular if every color induces a locally irregular subgraph of G. A colorable graph G is any graph which admits a locally irregular edge coloring. The locally irregular chromatic index X'irr(G) of a colorable graph G is the smallest number of colors required by a locally irregular edge coloring of G. The Local Irregularity Conjecture claims that all colorable graphs require at most 3 colors for a locally irregular edge coloring. Recently, it has been observed that the conjecture does not hold for the bow-tie graph B, since B is colorable and requires at least 4 colors for a locally irregular edge coloring. Since B is a cactus graph and all non-colorable graphs are also cacti, this seems to be a relevant class of graphs for the Local Irregularity Conjecture. In this paper we establish that X'irr(G)<= 4 for all colorable cactus graphs.

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