arXiv subjects

Mirko Petruševski

Publications and source records attributed to Mirko Petruševski.

10 recordsLinked to original sources

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

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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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

On Graph Odd Edge-Colorings and Odd Edge-Coverings

An odd $k$-edge-coloring of a graph $G$ is a (not necessarily proper) edge-coloring with at most $k$ colors such that each non-empty color class induces a graph in which every vertex is of odd degree; similarly, if more than one color per edge is allowed, we speak of an odd $k$-edge-covering of $G$. In this paper, we fully resolve two major conjectures on odd edge-colorings and odd edge-coverings of graphs, proposed by Petru{š}evski and {Š}krekovski ({\it European Journal of Combinatorics,} 91:103225, 2021). The first conjecture states that, apart from two particular exceptions which are respectively odd $5$- and odd-$6$-edge-colorable, for any other loopless and connected graph $G$ there exists an edge $e$ such that $G\backslash \{e\}$ is odd $3$-edge-colorable. The second conjecture states that any simple graph $G$ admits an odd $3$-edge-covering in which at most one edge receives more than one color. In addition, we strongly confirm the second conjecture by demonstrating that there exists an odd $3$-edge-covering in which at most one edge receives two colors and the rest of the edges receive unique colors.

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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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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

Colorings with neighborhood parity condition

In this short paper, we introduce a new vertex coloring whose motivation comes from our series on odd edge-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. We prove that every simple planar graph admits an odd $9$-coloring, and conjecture that $5$ colors always suffice.

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arXiv2022

On chromatic vertex stability of 3-chromatic graphs with maximum degree 4

The (independent) chromatic vertex stability ($\ivs(G)$) $\vs(G)$ is the minimum size of (independent) set $S\subseteq V(G)$ such that $χ(G-S)=χ(G)-1$. In this paper we construct infinitely many graphs $G$ with $Δ(G)=4$, $χ(G)=3$, $\ivs(G)=3$ and $\vs(G)=2$, which gives a partial negative answer to a problem posed in \cite{ABKM}.

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arXiv2022

Remarks on proper conflict-free colorings of graphs

A vertex coloring of a graph is said to be \textit{conflict-free} with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al., \textit{Proper Conflict-free and Unique-maximum Colorings of Planar Graphs with Respect to Neighborhoods}, arXiv preprint], the minimum number of colors in any such proper coloring of graph $G$ is the PCF chromatic number of $G$, denoted $χ_{\mathrm{pcf}}(G)$. In this paper, we determine the value of this graph parameter for several basic graph classes including trees, cycles, hypercubes and subdivisions of complete graphs. We also give upper bounds on $χ_{\mathrm{pcf}}(G)$ in terms of other graph parameters. In particular, we show that $χ_{\mathrm{pcf}}(G) \leq5Δ(G)/2$ and characterize equality. Several sufficient conditions for PCF $k$-colorability of graphs are established for $4\le k\le 6$. The paper concludes with few open problems.

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