Mirko Petrusevski - Academia.edu (original) (raw)

Papers by Mirko Petrusevski

Discrete mathematics letters, Jun 18, 2022

The (independent) chromatic vertex stability (ivsχ(G)) vsχ(G) is the minimum size of (independent... more The (independent) chromatic vertex stability (ivsχ(G)) vsχ(G) is the minimum size of (independent) set S ⊆ V (G) such that χ(G − S) = χ(G) − 1. The question of how large must the chromatic number χ(G) of a graph G be, in terms of the maximum degree ∆(G), to ensure the equality ivsχ(G) = vsχ(G) was raised by Akbari et al. [European J. Combin. 102 (2022) #103504]; the authors showed that ivsχ(G) = vsχ(G) if χ(G) ∈ {∆(G), ∆(G) + 1}, and also pointed out to graphs with χ(G) ≤ (∆(G) + 1)/2 for which ivsχ(G) > vsχ(G). In the light of their findings, they raised the following problem: Is it true that χ(G) ≥ ∆(G)/2 + 1 always implies ivsχ(G) = vsχ(G)? This threshold question was recently answered in the negative by Cambrie et al. [arXiv: 2203.13833v1, (2022)]. In this paper, we show that the smallest instance for counterexamples is the case (χ(G), ∆(G)) = (3, 4), with the smallest possible order being 9 (and there are 30 such graphs). We construct exponentially many graphs G having ∆(G) = 4, χ(G) = 3, ivsχ(G) = 3, and vsχ(G) = 2.

Discrete Applied Mathematics, Nov 1, 2022

In this short paper, we introduce a new vertex coloring whose motivation comes from our series on... more 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∈ V(G) there exists a color c such that φ^-1(c)∩ N(x) is odd-sized. We prove that every simple planar graph admits an odd 9-coloring, and conjecture that 5 colors always suffice.

Journal of Graph Theory, Feb 25, 2019

An odd graph is a graph whose vertex degrees are all odd. As introduced by Pyber in 1991, an odd ... more An odd graph is a graph whose vertex degrees are all odd. As introduced by Pyber in 1991, an odd edge-covering of graph G is a family of odd subgraphs that cover its edges.

Mathematics, Jan 18, 2021

Discrete Applied Mathematics

Ars Mathematica Contemporanea, 2021

Journal of Graph Theory, May 29, 2023

An odd graph is a finite graph all of whose vertices have odd degrees. Given graph G is decomposa... more 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 ≥ χ 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 ≤ 2. 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) ≤ 4. 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 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 ∈ 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.

Journal of Combinatorial Optimization, 2017

A vertex signature \pi π of a finite graph G is any mapping \pi \,{:}\,V(G)\rightarrow \{0,... more A vertex signature \pi π of a finite graph G is any mapping \pi \,{:}\,V(G)\rightarrow \{0,1\}$$π:V(G)→{0,1}. An edge-coloring of G is said to be vertex-parity for the pair (G,\pi )$$(G,π) if for every vertex v each color used on the edges incident to v appears in parity accordance with \pi π, i.e. an even or odd number of times depending on whether \pi (v)$$π(v) equals 0 or 1, respectively. The minimum number of colors for which (G,\pi )$$(G,π) admits such an edge-coloring is denoted by \chi '_p(G,\pi )$$χp′(G,π). We characterize the existence and prove that \chi '_p(G,\pi )$$χp′(G,π) is at most 6. Furthermore, we give a structural characterization of the pairs (G,\pi )$$(G,π) for which \chi '_p(G,\pi )=5$$χp′(G,π)=5 and \chi '_p(G,\pi )=6$$χp′(G,π)=6. In the last part of the paper, we consider a weaker version of the coloring, where it suffices that at every vertex, at least one color appears in parity accordance with \pi π. We show that the corresponding chromatic index is at most 3 and give a complete characterization for it.

Discrete Mathematics, 2016

We conjecture that every planar graph of odd-girth at least 11 admits a homomorphism to the Coxet... more We conjecture that every planar graph of odd-girth at least 11 admits a homomorphism to the Coxeter graph. Supporting this conjecture, we prove that every planar graph of odd-girth at least 17 admits a homomorphism to the Coxeter graph.

MATCH Communications in Mathematical and in Computer Chemistry

For a simple graph G with n vertices and m edges, the inequality M1(G)/n ≤ M2(G)/m, whereM1(G) an... more For a simple graph G with n vertices and m edges, the inequality M1(G)/n ≤ M2(G)/m, whereM1(G) andM2(G) are the first and the second Zagreb indices of G, is known as Zagreb indices inequality. Generalization of these indices gives first λM1(G) and second λM2(G) variable Zagreb indices. Vukiˇcevi´c in [13] has given an incomplete proof for the claim: for all simple graphs and all λ ∈ [0, 1 2 ], holds λM1(G)/n ≤λ M2(G)/m. Here we present a complete proof using Karamata’s inequality.

An edge coloring of a graph G is said to be an odd edge coloring if for each vertex v of G and ea... more An edge coloring of a graph G is said to be an odd edge coloring if for each vertex v of G and each color c, the vertex v uses the color c an odd number of times or does not use it at all. In [5], Pyber proved that 4 colors suffice for an odd edge coloring of any simple graph. Recently, some results on this type of colorings of (multi)graphs were successfully applied in solving a problem of facial parity edge coloring [3, 2]. In this paper we present additional results, namely we prove that 6 colors suffice for an odd edge coloring of any loopless connected (multi)graph, provide examples showing that this upper bound is sharp and characterize the family of loopless connected (multi)graphs for which the bound 6 is achieved. We also pose several open problems.

Mathematics

The principal aim of this article is to initiate a study of the following coloring notion for dig... more The principal aim of this article is to initiate a study of the following coloring notion for digraphs. An odd k-edge coloring of a general digraph (directed pseudograph) D is a (not necessarily proper) coloring of its edges with at most k colors such that for every vertex v and color c holds: if c is used on the set ∂D(v) of edges incident with v, then c appears an odd number of times on each non-empty set from the pair ∂D+(v),∂D−(v) of, respectively, outgoing and incoming edges incident with v. We show that it can be decided in polynomial time whether D admits an odd 2-edge coloring. Throughout the paper, several conjectures, questions and open problems are posed. In particular, we conjecture that for each odd edge-colorable digraph four colors suffice.

European Journal of Combinatorics

Ars Mathematica Contemporanea

The acyclic number a(G) of a graph G is the maximum order of an induced forest in G. The purpose ... more The acyclic number a(G) of a graph G is the maximum order of an induced forest in G. The purpose of this short paper is to propose a conjecture that a(G) ≥ 1 − 3 2g n holds for every planar graph G of girth g and order n, which captures three known conjectures on the topic. In support of this conjecture, we prove a weaker result that a(G) ≥ 1 − 3 g n holds. In addition, we give a construction showing that the constant 3 2 from the conjecture cannot be decreased.

Математички билтен/BULLETIN MATHÉMATIQUE DE LA SOCIÉTÉ DES MATHÉMATICIENS DE LA RÉPUBLIQUE MACÉDOINE

A weak-odd edge-coloring of a digraph D is a (not necessarily proper) edge-coloring such that for... more A weak-odd edge-coloring of a digraph D is a (not necessarily proper) edge-coloring such that for each vertex v ∈ V (D) at least one color c satisfies the following requirement: if d + (v) > 0 then c appears an odd number of times on the outgoing edges at v; and if d − (v) > 0 then c appears an odd number of times on the ingoing edges at v. The minimum number of colors sufficient for a weak-odd edge-coloring of D is the weak-odd chromatic index, denoted χ wo (D). In this article we prove that χ wo (D) ≤ 3 for every digraph D, and show that this bound is sharp. We study when does a graph admit an orientation so that the obtained digraph is weak-odd 1-edge-colorable. We also prove that every graph admits an orientation for which the obtained digraph is weak-odd 2-edge-colorable.

Математички билтен/BULLETIN MATHÉMATIQUE DE LA SOCIÉTÉ DES MATHÉMATICIENS DE LA RÉPUBLIQUE MACÉDOINE

A graph is odd if all its vertices have odd degrees. A Shannon triangle is a loopless graph on th... more A graph is odd if all its vertices have odd degrees. A Shannon triangle is a loopless graph on three pairwise adjacent vertices. If the parities of the sizes of its bouquets (of parallel edges) are denoted by p, q, r in nonincreasing order, with 2 (resp. 1) denoting an even-sized (resp. odd-sized) bouquet, we then say the Shannon triangle is of type (p, q, r). The minimum number of odd subgraphs which cover its edges is p + q + r. For a Shannon triangle of type (2, 2, 2) (resp. (2, 2, 1)) this number equals 6 (resp. 5). We prove that, by excluding these two types of Shannon triangles, every other loopless connected graph admits an edge cover by four odd subgraphs. 2010 Mathematics Subject Classication. 05C15. Key words and phrases. Edge cover, odd subgraph, odd edge-coloring, Shannon triangle.

Discrete mathematics letters, Jun 18, 2022

The (independent) chromatic vertex stability (ivsχ(G)) vsχ(G) is the minimum size of (independent... more The (independent) chromatic vertex stability (ivsχ(G)) vsχ(G) is the minimum size of (independent) set S ⊆ V (G) such that χ(G − S) = χ(G) − 1. The question of how large must the chromatic number χ(G) of a graph G be, in terms of the maximum degree ∆(G), to ensure the equality ivsχ(G) = vsχ(G) was raised by Akbari et al. [European J. Combin. 102 (2022) #103504]; the authors showed that ivsχ(G) = vsχ(G) if χ(G) ∈ {∆(G), ∆(G) + 1}, and also pointed out to graphs with χ(G) ≤ (∆(G) + 1)/2 for which ivsχ(G) > vsχ(G). In the light of their findings, they raised the following problem: Is it true that χ(G) ≥ ∆(G)/2 + 1 always implies ivsχ(G) = vsχ(G)? This threshold question was recently answered in the negative by Cambrie et al. [arXiv: 2203.13833v1, (2022)]. In this paper, we show that the smallest instance for counterexamples is the case (χ(G), ∆(G)) = (3, 4), with the smallest possible order being 9 (and there are 30 such graphs). We construct exponentially many graphs G having ∆(G) = 4, χ(G) = 3, ivsχ(G) = 3, and vsχ(G) = 2.

Discrete Applied Mathematics, Nov 1, 2022

In this short paper, we introduce a new vertex coloring whose motivation comes from our series on... more 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∈ V(G) there exists a color c such that φ^-1(c)∩ N(x) is odd-sized. We prove that every simple planar graph admits an odd 9-coloring, and conjecture that 5 colors always suffice.

Journal of Graph Theory, Feb 25, 2019

An odd graph is a graph whose vertex degrees are all odd. As introduced by Pyber in 1991, an odd ... more An odd graph is a graph whose vertex degrees are all odd. As introduced by Pyber in 1991, an odd edge-covering of graph G is a family of odd subgraphs that cover its edges.

Mathematics, Jan 18, 2021

Discrete Applied Mathematics

Ars Mathematica Contemporanea, 2021

Journal of Graph Theory, May 29, 2023

An odd graph is a finite graph all of whose vertices have odd degrees. Given graph G is decomposa... more 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 ≥ χ 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 ≤ 2. 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) ≤ 4. 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 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 ∈ 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.

Journal of Combinatorial Optimization, 2017

A vertex signature \pi π of a finite graph G is any mapping \pi \,{:}\,V(G)\rightarrow \{0,... more A vertex signature \pi π of a finite graph G is any mapping \pi \,{:}\,V(G)\rightarrow \{0,1\}$$π:V(G)→{0,1}. An edge-coloring of G is said to be vertex-parity for the pair (G,\pi )$$(G,π) if for every vertex v each color used on the edges incident to v appears in parity accordance with \pi π, i.e. an even or odd number of times depending on whether \pi (v)$$π(v) equals 0 or 1, respectively. The minimum number of colors for which (G,\pi )$$(G,π) admits such an edge-coloring is denoted by \chi '_p(G,\pi )$$χp′(G,π). We characterize the existence and prove that \chi '_p(G,\pi )$$χp′(G,π) is at most 6. Furthermore, we give a structural characterization of the pairs (G,\pi )$$(G,π) for which \chi '_p(G,\pi )=5$$χp′(G,π)=5 and \chi '_p(G,\pi )=6$$χp′(G,π)=6. In the last part of the paper, we consider a weaker version of the coloring, where it suffices that at every vertex, at least one color appears in parity accordance with \pi π. We show that the corresponding chromatic index is at most 3 and give a complete characterization for it.

Discrete Mathematics, 2016

We conjecture that every planar graph of odd-girth at least 11 admits a homomorphism to the Coxet... more We conjecture that every planar graph of odd-girth at least 11 admits a homomorphism to the Coxeter graph. Supporting this conjecture, we prove that every planar graph of odd-girth at least 17 admits a homomorphism to the Coxeter graph.

MATCH Communications in Mathematical and in Computer Chemistry

For a simple graph G with n vertices and m edges, the inequality M1(G)/n ≤ M2(G)/m, whereM1(G) an... more For a simple graph G with n vertices and m edges, the inequality M1(G)/n ≤ M2(G)/m, whereM1(G) andM2(G) are the first and the second Zagreb indices of G, is known as Zagreb indices inequality. Generalization of these indices gives first λM1(G) and second λM2(G) variable Zagreb indices. Vukiˇcevi´c in [13] has given an incomplete proof for the claim: for all simple graphs and all λ ∈ [0, 1 2 ], holds λM1(G)/n ≤λ M2(G)/m. Here we present a complete proof using Karamata’s inequality.

An edge coloring of a graph G is said to be an odd edge coloring if for each vertex v of G and ea... more An edge coloring of a graph G is said to be an odd edge coloring if for each vertex v of G and each color c, the vertex v uses the color c an odd number of times or does not use it at all. In [5], Pyber proved that 4 colors suffice for an odd edge coloring of any simple graph. Recently, some results on this type of colorings of (multi)graphs were successfully applied in solving a problem of facial parity edge coloring [3, 2]. In this paper we present additional results, namely we prove that 6 colors suffice for an odd edge coloring of any loopless connected (multi)graph, provide examples showing that this upper bound is sharp and characterize the family of loopless connected (multi)graphs for which the bound 6 is achieved. We also pose several open problems.

Mathematics

The principal aim of this article is to initiate a study of the following coloring notion for dig... more The principal aim of this article is to initiate a study of the following coloring notion for digraphs. An odd k-edge coloring of a general digraph (directed pseudograph) D is a (not necessarily proper) coloring of its edges with at most k colors such that for every vertex v and color c holds: if c is used on the set ∂D(v) of edges incident with v, then c appears an odd number of times on each non-empty set from the pair ∂D+(v),∂D−(v) of, respectively, outgoing and incoming edges incident with v. We show that it can be decided in polynomial time whether D admits an odd 2-edge coloring. Throughout the paper, several conjectures, questions and open problems are posed. In particular, we conjecture that for each odd edge-colorable digraph four colors suffice.

European Journal of Combinatorics

Ars Mathematica Contemporanea

The acyclic number a(G) of a graph G is the maximum order of an induced forest in G. The purpose ... more The acyclic number a(G) of a graph G is the maximum order of an induced forest in G. The purpose of this short paper is to propose a conjecture that a(G) ≥ 1 − 3 2g n holds for every planar graph G of girth g and order n, which captures three known conjectures on the topic. In support of this conjecture, we prove a weaker result that a(G) ≥ 1 − 3 g n holds. In addition, we give a construction showing that the constant 3 2 from the conjecture cannot be decreased.

Математички билтен/BULLETIN MATHÉMATIQUE DE LA SOCIÉTÉ DES MATHÉMATICIENS DE LA RÉPUBLIQUE MACÉDOINE

A weak-odd edge-coloring of a digraph D is a (not necessarily proper) edge-coloring such that for... more A weak-odd edge-coloring of a digraph D is a (not necessarily proper) edge-coloring such that for each vertex v ∈ V (D) at least one color c satisfies the following requirement: if d + (v) > 0 then c appears an odd number of times on the outgoing edges at v; and if d − (v) > 0 then c appears an odd number of times on the ingoing edges at v. The minimum number of colors sufficient for a weak-odd edge-coloring of D is the weak-odd chromatic index, denoted χ wo (D). In this article we prove that χ wo (D) ≤ 3 for every digraph D, and show that this bound is sharp. We study when does a graph admit an orientation so that the obtained digraph is weak-odd 1-edge-colorable. We also prove that every graph admits an orientation for which the obtained digraph is weak-odd 2-edge-colorable.

Математички билтен/BULLETIN MATHÉMATIQUE DE LA SOCIÉTÉ DES MATHÉMATICIENS DE LA RÉPUBLIQUE MACÉDOINE

A graph is odd if all its vertices have odd degrees. A Shannon triangle is a loopless graph on th... more A graph is odd if all its vertices have odd degrees. A Shannon triangle is a loopless graph on three pairwise adjacent vertices. If the parities of the sizes of its bouquets (of parallel edges) are denoted by p, q, r in nonincreasing order, with 2 (resp. 1) denoting an even-sized (resp. odd-sized) bouquet, we then say the Shannon triangle is of type (p, q, r). The minimum number of odd subgraphs which cover its edges is p + q + r. For a Shannon triangle of type (2, 2, 2) (resp. (2, 2, 1)) this number equals 6 (resp. 5). We prove that, by excluding these two types of Shannon triangles, every other loopless connected graph admits an edge cover by four odd subgraphs. 2010 Mathematics Subject Classication. 05C15. Key words and phrases. Edge cover, odd subgraph, odd edge-coloring, Shannon triangle.