# Roman SotákPavol Jozef Šafárik University in Košice · Institute of Mathematics

Roman Soták

Assoc. Prof., Dr., PhD.

## About

69

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620

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Introduction

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January 2010 - present

## Publications

Publications (69)

A matching $M$ in a graph $G$ is {\em semistrong} if every edge of $M$ has an endvertex of degree one in the subgraph induced by the vertices of $M$. A {\em semistrong edge-coloring} of a graph $G$ is a proper edge-coloring in which every color class induces a semistrong matching. In this paper, we continue investigation of properties of semistrong...

The weight of an edge is the degree‐sum of its end‐vertices. An edge u v $uv$ is an ( i , j ) $(i,j)$‐edge if deg ( u ) ≤ i $\,\text{deg}\,(u)\le i$ and deg ( v ) ≤ j $\,\text{deg}\,(v)\le j$. In 1955, Kotzig proved that every 3‐connected plane graph contains an edge of weight at most 13. Later, Borodin proved the existence of such an edge in plane...

A {\em conflict-free coloring} of a graph {\em with respect to open} (resp., {\em closed}) {\em neighborhood} is a coloring of vertices such that for every vertex there is a color appearing exactly once in its open (resp., closed) neighborhood. Similarly, a {\em unique-maximum coloring} of a graph {\em with respect to open} (resp., {\em closed}) {\...

A proper edge coloring of a graph is strong if it creates no bichromatic path of length three. It is well known that for a strong edge coloring of a $k$-regular graph at least $2k-1$ colors are needed. We show that a $k$-regular graph admits a strong edge coloring with $2k-1$ colors if and only if it covers the Kneser graph $K(2k-1,k-1)$. In partic...

The weight of an edge e is the degree-sum of its end-vertices. An edge e=uv is an (i,j)-edge if deg(u)≤i and deg(v)≤j. In 1955, Kotzig proved that every 3-connected planar graph contains an edge of weight at most 13. Later, Borodin extended this result to the class of simple planar graphs with minimum degree at least 3. If we consider the class of...

A star edge-coloring of a graph G is a proper edge-coloring without bichromatic paths or cycles of length four. The smallest integer k such that G admits a star edge-coloring with k colors is the star chromatic index of G. In the seminal paper on the topic, Dvo řák, Mohar, and Šámal asked if the star chromatic index of complete graphs is linear in...

A cyclic coloring of a plane graph G is a coloring of its vertices such that vertices incident with the same face have distinct colors. The minimum number of colors in a cyclic coloring of a plane graph G is its cyclic chromatic number χc(G). Let Δ∗(G) be the maximum face degree of a graph G.
In this note we show that to prove the Cyclic Coloring C...

A proper edge coloring of a graph is strong if it creates no bichromatic path of length three. It is well known that for a strong edge coloring of a $k$-regular graph at least $2k-1$ colors are needed. We show that a $k$-regular graph admits a strong edge coloring with $2k-1$ colors if and only if it covers the Kneser graph $K(2k-1,k-1)$. In partic...

A cyclic coloring of a plane graph $G$ is a coloring of its vertices such that vertices incident with the same face have distinct colors. The minimum number of colors in a cyclic coloring of a plane graph $G$ is its cyclic chromatic number $\chi_c(G)$. Let $\Delta^*(G)$ be the maximum face degree of a graph $G$. In this note we show that to prove t...

A \textit{$d$-subsequence} of a sequence $\varphi = x_1\dots x_n$ is a subsequence $x_i x_{i+d} x_{i+2d} \dots$, for any positive integer $d$ and any $i$, $1 \le i \le n$. A \textit{$k$-Thue sequence} is a sequence in which every $d$-subsequence, for $1 \le d \le k$, is non-repetitive, i.e. it contains no consecutive equal subsequences. In 2002, Gr...

A star edge-coloring of a graph $G$ is a proper edge-coloring without bichromatic paths and cycles of length four. The smallest integer $k$ such that $G$ admits a star edge-coloring with $k$ colors is the star chromatic index of $G$. In the seminal paper on the topic, Dvo\v{r}\'{a}k, Mohar, and \v{S}\'{a}mal asked if the star chromatic index of com...

A graph is 1‐planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal 1‐planar. For a 3‐connected locally maximal 1‐planar...

A graph is $1$-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal $1$-planar. For a $3$-connected locally maximal $1...

Deciding whether a planar graph (even of maximum degree 4) is 3-colorable is NP-complete. Determining subclasses of planar graphs being 3-colorable has a long history, but since Grötzsch’s result that triangle-free planar graphs are such, most of the effort was focused to solving Havel’s and Steinberg’s conjectures. In this paper, we prove that eve...

A graph is \textit{locally irregular} if the neighbors of every vertex $v$ have degrees distinct from the degree of $v$. A \textit{locally irregular edge-coloring} of a graph $G$ is an (improper) edge-coloring such that the graph induced on the edges of any color class is locally irregular. It is conjectured that $3$ colors suffice for a locally ir...

A $d$-subsequence of a sequence $\varphi = x_1\dots x_n$ is a subsequence $x_i x_{i+d} x_{i+2d} \dots$, for any positive integer $d$ and any $i$, $1 \le i \le n$. A \textit{$k$-Thue sequence} is a sequence in which every $d$-subsequence, for $1 \le d \le k$, is non-repetitive, i.e. it contains no consecutive equal subsequences. In 2002, Grytczuk pr...

Deciding whether a planar graph (even of maximum degree $4$) is $3$-colorable is NP-complete. Determining subclasses of planar graphs being $3$-colorable has a long history, but since Gr\"{o}tzsch's result that triangle-free planar graphs are such, most of the effort was focused to solving Havel's and Steinberg's conjectures. In this paper, we prov...

A presentatiton at the Eight Cracow Conference on Graph Theory "Rytro '18".

{\emph A star edge-coloring} of a graph is a proper edge-coloring without bichromatic paths and cycles of length four. In this paper, we consider the list version of this coloring and prove that the list star chromatic index of every subcubic graph is at most $7$, answering the question of Dvo\v{r}\'{a}k et al. (Star chromatic index, J. Graph Theor...

Let G be a cellular embedding of a multigraph in a 2-manifold. Two distinct edges e1, e2 ∈ E(G) are facially adjacent if they are consecutive on a facial walk of a face f ∈ F (G). An incidence of the multigraph G is a pair (v, e), where v ∈ V (G), e ∈ E(G) and v is incident with e in G. Two distinct incidences (v1, e1) and (v2, e2) of G are faciall...

An incidence in a graph G is a pair (v, e) where v is a vertex of G and e is an edge of G incident to v. Two incidences (v, e) and (u, f) are adjacent if at least one of the following holds: (i) v = u, (ii) e = f, or (iii) edge vu is from the set {e, f}. An incidence coloring of G is a coloring of its incidences assigning distinct colors to adjacen...

An edge-coloring of a graph $G$ can be viewed as an edge-decomposition, where the edges of each color class represent a subgraph $H$ of $G$. Different types of edge-colorings induce different subgraphs. In an ordinary proper edge-coloring, for example, every subgraph $H$ is a matching. We discuss several types of edge-colorings, each having the pro...

A star edge-coloring of a graph is a proper edge-coloring without bichromatic paths and cycles of length four. We consider the list version of this coloring and prove that the list star chromatic index of every subcubic graph is at most 7, answering the question of Dvořák et al. in [Dvořák, Z., B. Mohar, and R. Šámal, Star chromatic index, J. Graph...

An incidence in a graph G is a pair (v, e) where v is a vertex of G and e is an edge of G incident to v. Two incidences (v, e) and (u, f) are adjacent if at least one of the following holds: (a)v=u,(b)e=f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage...

A sequence r1,r2,...,r2n is called an anagram if rn+1,rn+2,...,r2n is a permutation of r1,r2,...,rn. A sequence S is called anagram-free if no block (i.e. subsequence of consecutive terms of S) is an anagram. A coloring of the edges of a given plane graph G is called facial anagram-free if the sequence of colors on any facial trail (i.e. a trail of...

An NMNR-coloring of a hypergraph is a coloring of vertices such that in every hy-peredge at least two vertices are colored with distinct colors, and at least two vertices are colored with the same color. We prove that every 3-uniform 3-regular hypergraph admits an NMNR-coloring with at most 3 colors. As a corollary, we confirm the conjecture that e...

An \textit{incidence} in a graph $G$ is a pair $(v,e)$ where $v$ is a vertex of $G$ and $e$ is an edge of $G$ incident to $v$. Two incidences $(v,e)$ and $(u,f)$ are \textit{adjacent} if at least one of the following holds: $(a)$ $v = u$, $(b)$ $e = f$, or $(c)$ $vu \in \{e,f\}$. An \textit{incidence coloring} of $G$ is a coloring of its incidences...

In this paper we study unavoidable sets of types of 3-paths for families of planar graphs with minimum degree at least 2 and a given girth g. A 3-path of type (i,j,k) is a path uvw on three vertices u, v, and w such that the degree of u (resp. v, resp. w) is at most i (resp. j, resp. k). The elements i,j,k are called parameters of the type. The set...

A star edge coloring of a graph is a proper edge coloring without
bichromatic paths and cycles of length four. In this paper we establish tight
upper bounds for trees and subcubic outerplanar graphs, and derive an upper
bound for outerplanar graphs.

For an assignment of numbers to the vertices of a graph, let S[u] be the sum of the labels of all the vertices in the closed neighborhood of u, for a vertex u. Such an assignment is called closed distinguishing if S[u]≠S[v] for any two adjacent vertices u and v unless the closed neighborhoods of u and v coincide. In this note we investigate dis[G],...

An (i, j, k)-path is a path on three vertices u, v and w in this order with deg(u) <= i, deg(v) <= j, and deg(w) <= k. In this paper, we prove that every connected plane graph of girth 4 and minimum degree at least 2 has at least one of the following: a (2, infinity, 2)-path, a (2, 7, 3)-path, a (3, 5, 3)-path, a (4, 2, 5)-path, or a (4, 3, 4)-path...

A graph is called fractional (r/s, d) -defective colorable if its vertices can be colored with r colors in such a way that each vertex receives s distinct colors and has at most d defects (a defect corresponds to the situation when two adjacent vertices are assigned with non-disjoint sets). We show that each outerplanar graph having no triangle fac...

Let P and Q be additive and hereditary graph properties, r, s ∈ ℕ, r ≥ s, and [ℤr]s be the set of all s-element subsets of ℤr. An (r, s)-fractional (P, Q)-total coloring of G is an assignment h: V(G) U E(G) → [ℤr]s such that for each i G ℤr the following holds: the vertices of G whose color sets contain color i induce a subgraph of G with property...

Let P and Q be additive and hereditary graph properties and let r, s be integers such that r ≥ s. Then an r/s-fractional (P, Q)-total coloring of a finite graph G = (V, E) is a mapping f, which assigns an s-element subset of the set {1, 2,⋯, r} to each vertex and each edge, moreover, for any color i all vertices of color i induce a subgraph with pr...

A sequence is Thue or nonrepetitive if it does not contain a repetition of any length. We consider a generalization of this notion. A j-subsequence of a sequence S is a subsequence in which two consecutive terms are at indices of difference j in S. A k-Thue sequence is a sequence in which every j-subsequence, for 1 <= j <= k, is also Thue. It was c...

For a given graph H and n≥1, let f(n,H) denote the maximum number m for which it is possible to colour the edges of the complete graph Kn with m colours in such a way that each subgraph H in Kn has at least two edges of the same colour. Equivalently, any edge-colouring of Kn with at least rb(n,H)=f(n,H)+1 colours contains a rainbow copy of H. The n...

In this paper we study the existence of unavoidable paths on three vertices in sparse graphs. A path uvw on three vertices u, v, and w is of type (i, j, k) if the degree of u, (respectively v, w) is at most i (respectively j, k) we that every graph with minimum degree at least 2 and average degree strictly less than in contains a path of one of the...

An l-facial edge coloring of a plane graph is a coloring of the edges such
that any two edges at distance at most l on a boundary walk of some face
receive distinct colors. It is conjectured that 3l + 1 colors suffice for an
l-facial edge coloring of any plane graph. We prove that 7 colors suffice for a
2-facial edge coloring of any plane graph and...

Given an integer valued weighting of all elements of a 2-connected plane graph G with vertex set V, let c(v) denote the sum of the weight of v is an element of V and of the weights of all edges and all faces incident with v. This vertex coloring of G is proper provided that c(u) not equal c(v) for any two adjacent vertices u and v of G. We show tha...

The coloring of disk graphs is motivated by the frequency assignment problem. In 1998, Malesińska et al. introduced double disk graphs as their generalization. They showed that the chromatic number of a double disk graph \(G\) is at most \(33\,\omega (G) - 35\), where \(\omega (G)\) denotes the size of a maximum clique in \(G\). Du et al. improved...

A strong edge-coloring of a graph is a proper edge-coloring where the edges
at distance at most two receive distinct colors. It is known that every planar
graph with maximum degree D has a strong edge-coloring with at most 4D + 4
colors. We show that 3D + 6 colors suffice if the graph has girth 6, and 3D
colors suffice if the girth is at least 7. M...

Graph Theory
International audience
For a positive integer n∈ℕ and a set D⊆ ℕ, the distance graph GnD has vertex set { 0,1,\textellipsis,n-1} and two vertices i and j of GnD are adjacent exactly if |j-i|∈D. The condition gcd(D)=1 is necessary for a distance graph GnD being connected. Let D={d1,d2}⊆ℕ be such that d1>d2 and gcd(d1,d2)=1. We prove the...

A strong edge coloring of a graph $G$ is a proper edge coloring in which each
color class is an induced matching of $G$. In 1993, Brualdi and Quinn Massey
proposed a conjecture that every bipartite graph without $4$-cycles and with
the maximum degrees of the two partite sets $2$ and $\Delta$ admits a strong
edge coloring with at most $\Delta+2$ col...

Let φ:E→{1,2,⋯,k} be a proper edge coloring of a graph G=(V,E). The set of colors of edges incident to a vertex v∈V is called the color set of v and denoted by S(v). The coloring φ is vertex-distinguishing if S(u)≠S(v) for any two distinct vertices u,v∈V. A d-strong edge coloring of a graph G is a proper edge coloring that distinguishes any two dis...

For a given graph H and n ? 1; let f(n;H) denote the maximum number m for
which it is possible to colour the edges of the complete graph Kn with m
colours in such a way that each subgraph H in Kn has at least two edges of the
same colour. Equivalently, any edge-colouring of Kn with at least rb(n;H) =
f(n;H)+1 colours contains a rainbow copy of H: T...

A facial parity edge colouring of a connected bridgeless plane graph is such an edge colouring in which no two face-adjacent edges receive the same colour and, in addition, for each face ff and each colour cc, either no edge or an odd number of edges incident with ff is coloured with cc. Let χp′(G) denote the minimum number of colours used in such...

An acyclic edge coloring of a graph is a proper edge coloring without bichromatic cycles. In 1978, it was conjectured that Δ ( G ) + 2 colors suffice for an acyclic edge coloring of every graph G (Fiamčík, 1978 [8]). The conjecture has been verified for several classes of graphs, however, the best known upper bound for as special class as planar gr...

Let P be a graph property and r, s ε ℕ, r > s. A strong circular (P, r, s)-colouring of a graph G is an assignment f : V(G) → {0,1,...,r - 1}, such that the edges uv G E(G) satisfying |f(u) - f (v)| < s or |f (u) - f (v)| > r - s, induce a subgraph of G with the propery P. In this paper we present some basic results on strong circular (P, r, s)-col...

A sequence r1, r2, …, r2n such that ri=rn+ i for all 1≤i≤n is called a repetition. A sequence S is called non-repetitive if no block (i.e. subsequence of consecutive terms of S) is a repetition. Let G be a graph whose edges are colored. A trail is called non-repetitive if the sequence of colors of its edges is non-repetitive. If G is a plane graph,...

An edge colouring of a graph G without isolated edges is neighbour-distinguishing if any two adjacent vertices have distinct sets consisting of colours of their incident edges. The general neighbour-distinguishing index of G is the minimum number gndi(G) of colours in a neighbour-distinguishing edge colouring of G. Győri et al. [E. Győri, M. Horňák...

A graph H is said to be light in a family X of graphs if each graph G is an element of H containing a subgraph isomorphic to H contains also an isomorphic copy of H such that each its vertex has the degree (in G) bounded above by a finite number phi(H, H) depending only on H and H. We prove that in the family of all 3-connected plane graphs of mini...

A face of an edge-colored plane graph is called rainbow if the number of colors used on its edges is equal to its size. The maximum number of colors used in an edge coloring of a connected plane graph Gwith no rainbow face is called the edge-rainbowness of G. In this paper we prove that the edge-rainbowness of Gequals the maximum number of edges of...

The vertex-distinguishing index χs′(G) of a graph G is the minimum number of colours required to properly colour the edges of G in such a way that any two vertices are incident with different sets of colours. We consider this parameter for some regular graphs. Moreover, we prove that for any graph, the value of this invariant is not changed if the...

Deza and Grishukhin studied 3-valent maps Mn(p, q) consisting of a ring of n q-gons whose inner and outer domains are filled by p-gons. They described the conditions for n, p, q under which such map may exist and presented several infinite families of them. We extend their results by presenting several new maps concerning mainly large values of n a...

A total edge irregular k-labelling ν of a graph G is a labelling of the vertices and edges of G with labels from the set {1,…,k} in such a way that for any two different edges e and f their weights φ(f) and φ(e) are distinct. Here, the weight of an edge g=uv is φ(g)=ν(g)+ν(u)+ν(v), i. e. the sum of the label of g and the labels of vertices u and v....

A k-ranking of a graph G = (V, E) is a mapping ϕ: V → {1, 2, ..., k} such that each path with end vertices of the same colour c contains an internal vertex with colour greater than c. The ranking number of a graph G is the smallest positive integer k admitting a k-ranking of G. In the on-line version of the problem, the vertices v
1, v
2, ..., v
n...

For a given simple graph an average labelling is defined. The graphs with average labellings and all the admissible average labellings for such graphs are characterized.

A subgraph of a plane graph is light if each of its vertices has a small degree in the entire graph. Consider the class T(5) of plane triangulations of minimum degree 5. It is known that each G ϵ T(5) contains a light triangle. From a recent result of Jendrol' and Madaras the existence of light cycles C4 and C5 in each G ϵ T(5) follows. We prove he...

A subgraph of a plane graph is light if each of its vertices has a small degree in entire graph. Consider the class $T (5)$ of all plane triangulations of minimum degree $5$. It is known that each $G \in T (5)$ contains a light triangle. From a recent result of Jendrol and Madaras, the existence of light cycles $C4$ and $C5$ in each $G \in T (5)$ f...

For given nonnegative integers k, s an upper bound on the minimum number of vertices of a strongly connected digraph with exactly k kernels and s solutions is presented.

Integers have many interesting properties. In this paper it will be shown that, for an arbitrary nonconstant arithmetic progression {a n }™ =l of positive integers (denoted by N), either {a n }™ =l contains infinitely many palindromic numbers or else 10|a w for every n GN. (This result is a generalization of the theorem concerning the existence of...

Observability of a graph G is the minimum k for which the edges of G can be properly coloured with k colours in such a way that colour sets of vertices of G (sets of colours of their incident edges) are pairwise distinct. It is shown that limn→∞obs(Qn)n = 1 + q∗ where q∗ = 0.293815 … is the unique solution of the equation (x + 1)x+1 = 2xx in the in...

The point-distinguishing chromatic index of a graph represents the minimum number of colours in its edge colouring such that each vertex is distinguished by the set of colours of edges incident with it. Asymptotic information on jumps of the point-distinguishing chromatic index of K n,n is found.

The point-distinguishing chromatic index x0(G) of a graph G represents the minimum number of colours in an edge colouring of G such that each vertex of G is distinguished by the set of colours of its incident edges. It is known that x0(Kn,n) is a non-decreasing function of n with jumps of value 1. We prove that x0(K46,46) = 7 and x0(K47,47) = 8.

Observability, a graph invariant inspired by the point-distinguishing chromatic index, the line-distinguishing chromatic number and the harmonious chromatic number, is introduced. Its value has been determined for complete graphs, paths, cycles, wheels and complete bipartite graphs. Special attention is devoted to regular graphs with optimum struct...

The degree profile of an edge e of a finite hypergraph H is the map assigning to a positive integer i the number of vertices of degree i incident with e. The edge degree profile of H is the map describing for any possible degree profile α the number of edges in H with degree profile α. A necessary and sufficient condition for existence of hypergrap...

## Projects

Project (1)

Adding additional (or different) assumptions to an edge coloring can result in combinatorial problems which need an entirely different approach to solve as the usual (classical) coloring problems. In the scope of this project we are investigating various edge-coloring types.
Some brief and compact collection of considered edge-coloings may be found here: https://prezi.com/sxe0x9uopj30/edge-coloring/?utm_campaign=share&utm_medium=copy