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Riste Škrekovski

Publications and source records attributed to Riste Škrekovski.

At least 19 recordsLinked to original sources
arXiv2026

Two conjectures on the cospectrality of complete tripartite graphs are false

For a graph G of order n with adjacency eigenvalues lambda_1(G) >= ... >= lambda_n(G), the cospectrality of G within the set G of all graphs of order n is the smallest spectral distance from G to another graph of G. Abdollahi and Zakeri conjectured that for every n >= 4 the cospectrality of the complete tripartite graph K_{n,n,n}, both in the Manhattan and in the Euclidean spectral distance, is attained precisely at K_{n-1,n,n+1}. We show that both conjectures are false for every n >= 4, that is, in the whole range in which they are stated. The counterexample is the graph H_n obtained from K_{n-1,n,n} by adding a vertex adjacent to the part of cardinality n - 1, and it is the natural generalization of the graph which the authors themselves exhibited for n = 3 and considered to be exceptional. We determine the spectrum of H_n in a closed form, and we show that the two spectral distances tend to 1 and to 1/2 respectively, instead of to (2*sqrt(3))/3 and to 2/3 as conjectured. On the other hand, we show that in the Euclidean distance the conjecture becomes asymptotically true when the competing graphs are restricted to complete multipartite graphs, so that it fails because a graph of rank four beats the conjectured minimizer.

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arXiv2026

The rich-neighbor chromatic index is linear in the maximum degree

An edge of a multigraph is rich with respect to a proper edge-coloring if all the edges adjacent to it receive mutually distinct colors, and a proper edge-coloring is a rich-neighbor coloring if every non-isolated edge has a rich edge adjacent to it. The least number of colors in such a coloring is the rich-neighbor chromatic index chi'_rn(G). So far it was studied for subcubic graphs only, and multigraphs of larger maximum degree are considered here for the first time. The only upper bound valid in general was the quadratic one inherited from the strong chromatic index, while a single rich edge already requires 2D-1 colors in a D-regular graph, where D denotes the maximum degree. We give an upper linear bound chi'_rn(G) <= 6D-8 which holds for every loopless multigraph G with D >= 3, and we show that a coloring realizing this bound can be found in polynomial time. Consequently chi'_rn(G) = Theta(D). For D-regular graphs with D >= 3 the lower bound is 2D-1, and the upper bound is 5D-8. The bound 5D-8 holds more generally for every multigraph with D >= 3 which has a dominating P_{>=3}-system, and we characterize these multigraphs: such a system exists if and only if k_2(G-T) <= 2|T| holds for every set T of vertices, where k_2 counts the components on two vertices. In particular the bound holds for every graph with 3 <= D <= 4d-4, where d denotes the minimum degree. The paper is concluded with a conjecture on the tight upper bound on chi'_rn(G).

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arXiv2026

Spreads of degrees in maximal planar graphs -- an exposition

For a graph $G$ and a set $B\subseteq V(G)$, the spread $\mathrm{sp}(B)$ of $B$ is the difference between the largest and the smallest degree in $G$ of a vertex of $B$, and for an integer $k\geq 0$ the parameter $\mathrm{sp}(G,k)$ is the largest cardinality of a set $B$ with $\mathrm{sp}(B)\leq k$. Caro, Lauri and Zarb asked for the minimum of $\mathrm{sp}(G,k)$ over the maximal planar graphs of order $n$; we write $\mathrm{MP}(n,δ,k)$ for this minimum over the maximal planar graphs of order $n$ and minimum degree $δ\in \{3,4,5\}$. This manuscript is intended as an exposition of the subject of spread in the degree sequence of a graph $G$, with emphasis on maximal planar graphs. We apply the general lower bound of the companion paper \cite{P1} to this class, together with further ideas, some based on Caro--West and Caro--Lauri--Zarb and some new. In particular we asymptotically determine the values of $\mathrm{MP}(n,δ,k)$ for every pair $(δ,k)$ with $δ\in \{3,4,5\}$ and $k\geq 0$. This solution recovers the case $\mathrm{MP}(n,δ,0)$, which is the repetition number $\mathrm{rep}(G)$ introduced by Caro and West.

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arXiv2026

On the odd independence number of the Queen graph

A set S of vertices of a graph is odd independent if it is independent and every vertex outside S has either zero or an odd number of neighbors in S. The largest size of such a set is the odd independence number alpha_od. Caro, Petrusevski, Skrekovski and Tuza [2] conjectured that alpha_od = 1 for every finite Queen graph. They also asked whether the infinite Queen graph has alpha_od = 1 or alpha_od = infinity. We prove that alpha_od = 1 in both cases. In particular, in the case of an infinite board we prove that alpha_od = 1 holds on the quarter plane and on the whole plane.

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arXiv2026

Resolutions of two conjectures on the spectral diameter

Let $λ_1(G) \geq \dots \geq λ_n(G)$ be the adjacency spectrum of a graph $G$ on $n$ vertices. The spectral distance $σ(G,H)$ between $n$-vertex graphs $G$ and $H$ is the Manhattan distance between their spectra, i.e. $σ(G,H) = \sum_{i=1}^n |λ_i(G) - λ_i(H)|$. Given a set $\mathcal{G}$ of pairwise non-isomorphic graphs of order $n$, the spectral diameter of $\mathcal{G}$ is defined as $\mathrm{sdiam}(\mathcal{G}) = \max\{\mathrm{secc}_{\mathcal{G}}(G) : G \in \mathcal{G}\}$, where $\mathrm{secc}_{\mathcal{G}}(G) = \max\{σ(G,H) : H \in \mathcal{G}\}$ is the spectral eccentricity of $G \in \mathcal{G}$. Among six conjectures on spectral distances posed by Z. Stanić in 2012, two conjectures related to the spectral diameter of certain graph classes remained open. One of them concerns the spectral diameter of the set $\mathcal{B}_n$ of all connected bipartite graphs of order $n$, while the other, of the set $\mathcal{T}_n$ of all trees of order $n$. More precisely, Stanić conjectured that $\mathrm{sdiam}(\mathcal{T}_n) = σ(P_n, K_{1,n-1})$, where $P_n$ is the path graph, while $K_{1,n-1}$ is the star, and that $\mathrm{sdiam}(\mathcal{B}_n) = \mathrm{secc}_{\mathcal{B}_n}(K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor})$, where $K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor}$ is the complete bipartite graph. In this paper, both of these conjectures are disproved.

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arXiv2026

A note on vertex-critical induced subgraphs of shift graphs

Shift graphs, introduced by Erdős and Hajnal in 1964, form one of the simplest known non-recursive constructions of triangle-free graphs with arbitrarily large chromatic number. In this note, we identify a surprising property: for each integer $k \geq 1$, the smallest $k$-chromatic shift graph contains a \emph{unique} induced $k$-vertex-critical subgraph. We give an explicit description of this subgraph and prove its uniqueness. This provides a new family of vertex-critical triangle-free graphs of arbitrarily large chromatic number.

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arXiv2026

An improved upper bound for the fair domination number of maximal outerplanar graphs

A dominating set $D$ of a graph $G$ is a \emph{fair dominating set} if every two vertices outside $D$ have the same number of neighbors in $D$, and the \emph{fair domination number} $\mathrm{fd}(G)$ is the minimum cardinality of such a set. Caro, Hansberg and Henning, who introduced this parameter, proved that $\mathrm{fd}(G)<17n/19$ for every maximal outerplanar graph $G$ of order $n\geq3$, and asked whether this bound is asymptotically best possible. We show that it is not the case by proving $\mathrm{fd}(G)\leq(7n-3)/8<7n/8$ for every maximal outerplanar graph $G$ of order $n\geq3$, and we exhibit an infinite family of maximal outerplanar graphs with $\mathrm{fd}(G)/n\rightarrow7/9$, so that the best asymptotic constant lies between $7/9$ and $7/8$.

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arXiv2026

Spreads of degrees in graphs

For a graph $G$ and a set $B\subseteq V(G)$, the spread $\mathrm{sp}(B)$ of $B$ is the difference between the largest and the smallest degree in $G$ of a vertex of $B$, and for an integer $k\geq0$ the parameter $\mathrm{sp}(G,k)$ is the largest cardinality of a set $B$ with $\mathrm{sp}(B)\leq k$. Caro, Lauri and Zarb derived a lower bound for $\mathrm{sp}(G,k)$ and, among several families of graphs, considered \[ \mathrm{MOP}(n,k)=\min \{\mathrm{sp}(G,k):G\text{ is a maximal outerplanar graph of order }n\} \] and determined $\mathrm{MOP}(n,k)$ up to an additive constant for every $k\not =2,$ leaving the case $k=2$ open, with the bounds $4n/9\leq \mathrm{MOP}(n,2)\leq (5n+19)/11$. We first prove a lower bound on $\mathrm{sp}(G,k)$ for an arbitrary graph $G$ in terms of its order $n$, its number of edges $m$ and its minimum degree $δ$. This lower bound contains the bounds of Caro, Lauri and Zarb and, for $k=0$, the bound $\mathrm{rep}(G)\geq \left\lceil n/(2d-2δ+1)\right\rceil $ of Caro and West, where $d=2m/n$. We determine when this lower bound is attained, exhibit explicit graphs attaining it, and show that it is exact for all graphs once $n\geq n_{0}(δ,k,d)$. We then apply the bound to maximal outerplanar graphs: adjusting the count to this class we prove \[ \mathrm{MOP}(n,2)\geq \left\lceil \frac{4n+10}{9}\right\rceil \qquad \text{for every }n\geq 14, \] with equality for $n\equiv 2\ (\mathrm{mod}\ 18)$, and $\mathrm{MOP}(n,2)=4n/9+O(1)$ for every $n$.

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arXiv2026

Counterexamples to two conjectures on (1, 2)-domination in cubic graphs

Let G be a cubic graph of order n. The induced cycles vertex number cind(G) is the largest size of a vertex set that induces a 2-regular subgraph of G. By gamma_1,2(G) we denote the (1,2)-domination number of G. Erves and Tepeh introduced the trilobite graphs T_n, which satisfy gamma_1,2(T_n) > cind(T_n). They stated two conjectures, the first of which says that every cubic graph G with cind(G) >= n/2 + 2 satisfies gamma_1,2(G) <= cind(G). The second says that a connected cubic graph G satisfies gamma_1,2(G) > cind(G) if and only if G is a trilobite. We show that both conjectures are false. A computer search finds counterexamples that are not trilobites already for n = 18, 20 and 22. We also construct an infinite family H(k) of order n = 20 + 4k. For every k >= 0 we prove that cind(H(k)) = n/2 + 2 and gamma_1,2(H(k)) = n/2 + 3. Hence both conjectures fail for infinitely many orders n.

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arXiv2026

A complete characterization of maximally σ-irregular trees with prescribed maximum degree

The sigma-irregularity of a graph G = (V, E) is defined as the sum, over all edges uv in E, of (d(u) - d(v))^2, where d(u) denotes the degree of vertex u. A tree on n vertices with maximum degree Delta is called maximal if it attains the greatest possible sigma-irregularity among all such trees. The maximal trees are already known for chemical trees (Delta <= 4) and for Delta = 5. In this paper, we characterize the maximal trees for every Delta >= 6 when n >= Delta(Delta - 1) + 1. We introduce three families of trees, T'{n,Delta}, T''{n,Delta}, and T'''{n,Delta}. All trees within the same family have the same sigma-irregularity, and we derive an explicit formula for the value attained by each family. Comparing these formulas determines which family has the greatest sigma-irregularity for given n and Delta. We then prove that a tree is maximal if and only if it belongs to a family attaining this greatest value. In contrast to the case Delta <= 5, the two natural candidate families T'{n,Delta} and T''{n,Delta} are not sufficient, since for every Delta >= 7 and every n congruent to 3 modulo Delta, the trees in T'''{n,Delta} have strictly greater sigma-irregularity.

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arXiv2026

Counting geodesic paths in graphs

A geodesic is a shortest path which connects a pair of vertices of a graph G. In this paper we define the geodesic subpath number gpn(G) of a graph G as the number of geodesics in G. The number of subtrees and subpaths are already studied in literature, but they are both large quantities. Hence, the geodesic subpath number which is related to these quantities but smaller than both, seems worthy of investigation. We first consider extremal graphs with respect to the geodesic subpath number among all connected graphs on n vertices. This number is minimized by the so called geodetic graphs, i.e. graphs in which each pair of vertices is connected by precisely one geodesic. As for the graphs which maximize the geodesic subpath number, we provide an upper bound on gpn(G) in terms of n and we further consider several graph families which might have a large gpn(G). Yet, their value of gpn(G) still does not attain the established bound, so narrowing the gap remains as an open problem. We also consider the class of cactus graphs on n vertices and k cycles and among them characterize extremal graphs with respect to this new invariant.

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arXiv2026

The odd independence number of graphs, I: Foundations and classical classes

An odd independent set $S$ in a graph $G=(V,E)$ is an independent set of vertices such that, for every vertex $v \in V \setminus S$, either $N(v) \cap S = \emptyset$ or $|N(v) \cap S| \equiv 1$ (mod 2), where $N(v)$ stands for the open neighborhood of $v$. The largest cardinality of odd independent sets of a graph $G$, denoted $α_{od}(G)$, is called the odd independence number of $G$. This new parameter is a natural companion to the recently introduced strong odd chromatic number. A proper vertex coloring of a graph $G$ is a strong odd coloring if, for every vertex $v \in V(G)$, each color used in the neighborhood of $v$ appears an odd number of times in $N(v)$. The minimum number of colors in a strong odd coloring of $G$ is denoted by $χ_{so}(G)$. A simple relation involving these two parameters and the order $|G|$ of $G$ is $α_{od}(G)\cdotχ_{so}(G) \geq |G|$, parallel to the same on chromatic number and independence number. We develop several basic inequalities concerning $α_{od}(G)$, and use already existing results on strong odd coloring, to derive lower bounds for odd independence in many families of graphs. We prove that $α_{od}(G) = α(G^2)$ holds for all claw-free graphs $G$, and apply this result to prove that determining $α_{od}(G)$ is in general NP-hard (and also when restricted to line graphs). We also present many results, using various techniques, concerning the odd independence number of cycles, paths, Moore graphs, Kneser graphs, the complete subdivision $S(K_n)$ of $K_n$, the half graphs $H_{n,n}$, and $K_p \Box K_q$. Further, we consider the odd independence number of the hypercube $Q_d$ and also of the complements of triangle-free graphs. Many open problems for future research are stated. Further related results can be found in part II of this work, arXiv: 2510.01897.

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arXiv2026

Degree-choosability of proper conflict-free list coloring of sparse graphs

Given a graph $G$ and a mapping $f:V(G) \to \mathbb{N}$, an $f$-list assignment of $G$ is a function that maps each $v \in V(G)$ to a set of at least $f(v)$ colors. For an $f$-list assignment $L$ of a graph $G$, a proper conflict-free $L$-coloring of $G$ is a proper coloring $ϕ$ of $G$ such that for every vertex $v \in V(G)$, $ϕ(v) \in L(v)$ and some appears precisely once in the neighborhood of $v$. We say that $G$ is proper conflict-free $f$-choosable if for every $f$-list assignment $L$ of $G$, there exists a proper conflict-free $L$-coloring of $G$. If $G$ is proper conflict-free $f$-choosable and there is a constant $k$ such that $f(v)= d_G(v)+k$ for every vertex $v$ of $G$, then we say $G$ is proper conflict-free $({\rm degree}+k)$-choosable. In this paper, we consider graphs with a bounded maximum average degree. We show that every graph with the maximum average degree less than $\frac{10}{3}$ is proper conflict-free $({\rm degree}+3)$-choosable, and that every graph with the maximum average degree less than $\frac{18}{7}$ is proper conflict-free $({\rm degree}+2)$-choosable. As a result, every planar graph with girth at least $5$ is proper conflict-free $({\rm degree}+3)$-choosable, and every planar graph with girth at least $9$ is proper conflict-free $({\rm degree}+2)$-choosable.

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arXiv2025

The odd independence number of graphs, II: Finite and infinite grids and chessboard graphs

An odd independent set $S$ in a graph $G=(V,E)$ is an independent set of vertices such that, for every vertex $v \in V \setminus S$, either $N(v) \cap S = \emptyset$ or $|N(v) \cap S| \equiv 1$ (mod 2), where $N(v)$ stands for the open neighborhood of $v$. The largest cardinality of odd independent sets of a graph $G$, denoted $α_{od}(G)$, is called the odd independence number of $G$. This new parameter is a natural companion to the recently introduced strong odd chromatic number. A proper vertex coloring of a graph $G$ is a strong odd coloring if, for every vertex $v \in V(G)$, each color used in the neighborhood of $v$ appears an odd number of times in $N(v)$. The minimum number of colors in a strong odd coloring of $G$ is denoted by $χ_{so}(G)$. A simple relation involving these two parameters and the order $|G|$ of $G$ is $α_{od}(G)\cdotχ_{so}(G) \geq |G|$, parallel to the same on chromatic number and independence number. In the present work, which is a companion to our first paper on the subject [The odd independence number of graphs, I: Foundations and classical classes], we focus on grid-like and chessboard-like graphs and compute or estimate their odd independence number and their strong odd chromatic number. Among the many results obtained, the following give the flavour of this paper: (1) $0.375 \leq \varrho_{od}(P_\infty \Box P_\infty) \leq 0.384615...$, where $\varrho_{od}(P_\infty \Box P_\infty)$ is the odd independence ratio. (2) $χ_{so}(G_d) = 3$ for all $d \geq 1$, where $G_d$ is the infinite $d$-dimensional grid. As a consequence, $\varrho_{od}(G_d) \geq 1/3$. (3) The $r$-King graph $G$ on $n^2$ vertices has $α_{od}(G) = \lceil n/(2r+1) \rceil^2$. Moreover, $χ_{so}(G) = (2r + 1)^2$ if $n \geq 2r + 1$, and $χ_{so}(G) = n^2$ if $n \leq 2r$. Many open problems are given for future research.

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arXiv2025

Remarks on proper conflict-free degree-choosability of graphs with prescribed degeneracy

A proper coloring $ϕ$ of $G$ is called a proper conflict-free coloring of $G$ if for every non-isolated vertex $v$ of $G$, there is a color $c$ such that $|ϕ^{-1}(c)\cap N_G(v)|=1$. As an analogy of degree-choosability of graphs, we introduced the notion of proper conflict-free $({\rm degree}+k)$-choosability of graphs. For a non-negative integer $k$, a graph $G$ is proper conflict-free $({\rm degree}+k)$-choosable if for any list assignment $L$ of $G$ with $|L(v)|\geq d_G(v)+k$ for every vertex $v\in V(G)$, $G$ admits a proper conflict-free coloring $ϕ$ such that $ϕ(v)\in L(v)$ for every vertex $v\in V(G)$. In this note, we first remark if a graph $G$ is $d$-degenerate, then $G$ is proper conflict-free $({\rm degree}+d+1)$-choosable. Furthermore, when $d=1$, we can reduce the number of colors by showing that every tree is proper conflict-free $({\rm degree}+1)$-choosable. This motivates us to state a question.

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arXiv2025

Proper conflict-free degree-choosability of outerplanar graphs

A proper coloring $ϕ$ of $G$ is called a proper conflict-free coloring of $G$ if for every non-isolated vertex $v$ of $G$, there is a color $c$ such that $|ϕ^{-1}(c)\cap N_G(v)|=1$. As an analogy to degree-choosability of graphs, the authors recently, in a previous paper, introduced the notion of proper conflict-free $({\rm degree}+k)$-choosability of graphs. For a non-negative integer $k$, a graph $G$ is proper conflict-free $({\rm degree}+k)$-choosable if for any list assignment $L$ of $G$ with $|L(v)|\geq d_G(v)+k$ for every vertex $v\in V(G)$, $G$ admits a proper conflict-free coloring $ϕ$ such that $ϕ(v)\in L(v)$ for every vertex $v\in V(G)$. In this paper, we show that every connected outerplanar graph other than the $5$-cycle is proper conflict-free $({\rm degree}+2)$-choosable. This bound is tight in the sense that there are infinitely many connected outerplanar graphs that are not proper conflict-free $({\rm degree}+1)$-choosable. We conclude the paper with two questions for further work.

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arXiv2025

Results on proper conflict-free list coloring of graphs

Given a graph $G$ and a mapping $f:V(G) \to \mathbb{N}$, an $f$-list assignment of $G$ is a function that maps each $v \in V(G)$ to a set of at least $f(v)$ colors. For an $f$-list assignment $L$ of a graph $G$, a proper conflict-free $L$-coloring of $G$ is a proper coloring $ϕ$ of $G$ such that $ϕ(v) \in L(v)$ for every vertex $v\in V(G)$ and $v$ has a color that appears precisely once at its neighborhood for every non-isolated vertex $v\in V(G)$. We say that $G$ is proper conflict-free $f$-choosable if for any $f$-list assignment $L$ of $G$, there exists a proper conflict-free $L$-coloring of $G$. For a non-negative integer $k$, we say that $G$ is \emph{proper conflict-free $({\rm degree}+k)$-choosable} if $G$ is proper conflict-free $f$-choosable where $f$ is a mapping with $f(v)= d_G(v)+k$ for every vertex $v\in V(G)$. Motivated by degree-choosability of graphs, we investigate the proper conflict-free $({\rm degree}+k)$-choosability of graphs, especially for cases $k=1,2,3$. As the 5-cycle is not proper conflict-free $({\rm degree}+2)$-choosable and it is the only such graph we know, it is possible that every connected graph other than the 5-cycle is proper conflict-free $({\rm degree}+2)$-choosable and thus every graph is proper conflict-free $({\rm degree}+3)$-choosable. To support these, we show that every connected graph with maximum degree at most 3 distinct from the 5-cycle is proper conflict-free $(\text{degree}+2)$-choosable, and that $S(G)$ is proper conflict-free $(\text{degree}+2)$-choosable for every graph $G$, where $S(G)$ is a graph obtained from $G$ by subdividing each edge once. Furthermore, by adapting the technique of DP-colorings, we prove that every graph with maximum degree at most $4$ is proper conflict-free $({\rm degree}+3)$-choosable.

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arXiv2025

The subpath number of cactus graphs

The subpath number of a graph G is defined as the total number of subpaths in G, and it is closely related to the number of subtrees, a well-studied topic in graph theory. This paper is a continuation of our previous paper [5], where we investigated the subpath number and identified extremal graphs within the classes of trees, unicyclic graphs, bipartite graphs, and cycle chains. Here, we focus on the subpath number of cactus graphs and characterize all maximal and minimal cacti with n vertices and k cycles. We prove that maximal cacti are cycle chains in which all interior cycles are triangles, while the two end-cycles differ in length by at most one. In contrast, minimal cacti consist of k triangles, all sharing a common vertex, with the remaining vertices forming a tree attached to this joint vertex. By comparing extremal cacti with respect to the subpath number to those that are extremal for the subtree number and the Wiener index, we demonstrate that the subpath number does not correlate with either of these quantities, as their corresponding extremal graphs differ.

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