arXiv subjects

Mika Olsen

Publications and source records attributed to Mika Olsen.

12 recordsLinked to original sources

arXiv2026

Upper chromatic number of generalized quadrangles

We can regard a generalized quadrangle as a hypergraph in which points and lines are identified with vertices and hyperedges, respectively. A vertex coloring is said to be rainbow on a hyperedge if all vertices contained in that hyperedge are assigned pairwise distinct colors; if no hyperedge is rainbow, the coloring is termed rainbow-free. For a given hypergraph $H$, the maximum integer $k$ for which there exists a rainbow-free $k$-coloring is called the upper chromatic number of $H$, and is denoted by $\overline{\chi}(H)$.

math.CO↗

arXiv2025

Redicoloring some classes of circulant tournaments

Given a digraph $D$ with no loops, the \textit{dicoloring graph} of $D$, denoted by $\mathcal{D}_k(D)$, is the graph whose vertices are the acyclic $k$-colorings of $D$ and two colorings are adjacent in $\mathcal{D}_k(D)$ if they differ in color on exactly one vertex. In this paper, we prove that there is no expression $\phi(\vec{\chi})$ in terms of the dichromatic number $\vec\chi$, such that the graph $\mathcal{D}_k(D)$ is connected for all graphs $D$ and integers $k\geq \phi(\vec\chi)$. We give conditions for the dicoloring graph of two infinite families of circulant tournaments to be connected, and we provide upper bounds for its diameter. In particular, for the Payley tournament $\vec{C}_{7}(1,2,4)$, also known as $ST_7$, we prove that $\mathcal{D}_k(\vec{C}_{7}(1,2,4))$ is connected and has diameter 8, for each $k\geq 3$.

math.CO↗

arXiv2025

Acyclic and complete coloring of digraphs with the minimum and maximum possible numbers of colors

The dichromatic and diachromatic numbers of a digraph are the minimum and maximum numbers of colors, respectively, in acyclic and complete colorings of the digraph. In this paper, we construct, for all $r \leq t$, non-symmetric digraphs with dichromatic number $r$ and diachromatic number $t$. Moreover, we discuss the existence of asymmetric digraphs with dichromatic number $ r $ and diachromatic number $ t \geq r $, establishing a quadratic upper bound $ b(r) \leq t $.

math.CO↗

arXiv2023

How to construct the symmetric cycle of length 5 using Hajós construction with an adapted Rank Genetic Algorithm

In 2020 Bang-Jensen et. al. generalized the Hajós join of two graphs to the class of digraphs and generalized several results for vertex colorings in digraphs. Although, as a consequence of these results, a digraph can be obtained by Hajós constructions (directed Hajós join and identifying non-adjacent vertices), determining the Hajós constructions to obtain the digraph is a complex problem. In particular, Bang-Jensen et al. posed the problem of determining the Hajós operations to construct the symmetric 5-cycle from the complete symmetric digraph of order 3 using only Hajós constructions. We successfully adapted a rank-based genetic algorithm to solve this problem by the introduction of innovative recombination and mutation operators from graph theory. The Hajós Join became the recombination operator and the identification of independent vertices became the mutation operator. In this way, we were able to obtain a sequence of only 16 Hajós operations to construct the symmetric cycle of order 5.

math.CO↗

arXiv2023

A procedure to obtain symmetric cycles of any odd length using directed Hajós constructions

The dichromatic number of a digraph $D$ is the minimum number of colors of a vertex coloring of $D$ such that $D$ has no monochromatic cycles. The Hajós join were recently extended to digraphs (using the dichromatic number) by J. Bang-Jensen et. al. and Hajós (directed) operations is a tool to obtain r-(di)chromatic (di)graphs. J. Bang-Jensen et. al. posed in 2020 the problem of how to obtain the symmetric cycle of length 5 from symmetric cycles of length 3. We recently solved this problem by applying a genetic algorithm. In this article, a procedure is presented to construct any odd symmetric cycle by applying directed Hajós operations to symmetric cycles of length 3, thus, generalizing the known construction of the symmetric cycle of length 5. In addition, this procedure is analyzed to determine its computational complexity.

math.CO↗

arXiv2022

On the harmonious chromatic number of graphs

The harmonious chromatic number of a graph $G$ is the minimum number of colors that can be assigned to the vertices of $G$ in a proper way such that any two distinct edges have different color pairs. This paper gives various results on harmonious chromatic number related to homomorphisms, incidence graphs of finite linear systems, and some circulant graphs.

math.CO↗

arXiv2022

Zykov sums of digraphs with diachromatic number equal to their harmonious number

The dichromatic number and the diachromatic number are generalizations of the chromatic number and the achromatic number for digraphs considering acyclic colorings. In this paper, we determine the diachromatic number of digraphs arising from the Zykov sum of digraphs that admit a complete $k$-coloring with $k=\tfrac{1+\sqrt{1+4m}}{2}$ for a suitable $m$. Consequently, the diachromatic number equals the harmonious number for every digraph in this family. In particular, we study the chromatic number, the diachromatic number, and the harmonious chromatic number of the Zykov sum of cycles.

math.CO↗

arXiv2021

The digrundy number of digraphs

We extend the Grundy number and the ochromatic number, parameters on graph colorings, to digraph colorings, we call them {\emph{digrundy number}} and {\emph{diochromatic number}}, respectively. First, we prove that for every digraph the diochromatic number equals the digrundy number (as it happen for graphs). Then, we prove the interpolation property and the Nordhaus-Gaddum relations for the digrundy number, and improve the Nordhaus-Gaddum relations for the dichromatic and diachromatic numbers bounded previously by the authors in [Electron. J. Combin. 25 (2018) no. 3, Paper {\#} 3.51, 17 pp.]

math.CO↗

arXiv2020

Conditions on the regularity of balanced $c$-partite tournaments for the existence of strong subtournaments with high minimum degree

We consider the following problem posed by Volkmann in 2007: How close to regular must a c-partite tournament be, to secure a strongly connected subtournament of order $c$? We give sufficient conditions on the regularity of balanced $c$-partite tournaments to assure the existence of strong maximal subtournament with minimum degree at least $\left\lfloor \frac{c-2}{4}\right\rfloor+1$. We obtain this result as an application of counting the number of subtournaments of order $c$ for which a vertex has minimum out-degree (resp. in-degree) at most $q\geq 0$.

math.CO↗

arXiv2019

Achromatic number, achromatic index and diachromatic number of circulant graphs and digraphs

In this paper, we determine the achromatic and diachromatic numbers of some circulant graphs and digraphs each one with two lengths and give bounds for other circulant graphs and digraphs with two lengths. In particular, for the achromatic number we state that $α(C_{16q^2+20q+7}(1,2))=8q+5$, and for the diachromatic number we state that $dac(\overrightarrow{C}_{32q^2+24q+5}(1,2))=8q+3$. In general, we give the lower bounds $α(C_{4q^2+aq+1}(1,a))\geq 4q+1$ and $dac(\overrightarrow{C}_{8q^2+2(a+4)q+a+3}(1,a))\geq 4q+3$ when $a$ is a non quadratic residue of $\mathbb{Z}_{4q+1}$ for graphs and $\mathbb{Z}_{4q+3}$ for digraphs, and the equality is attained, in both cases, for $a=3$. Finally, we determine the achromatic index for circulant graphs of $q^2+q+1$ vertices when the projective cyclic plane of odd order $q$ exists.

math.CO↗

arXiv2019

On the packing chromatic number of Moore graphs

The \emph{packing chromatic number $χ_ρ(G)$} of a graph $G$ is the smallest integer $k$ for which there exists a vertex coloring $Γ: V(G)\rightarrow \{1,2,\dots , k\}$ such that any two vertices of color $i$ are at distance at least $i + 1$. For $g\in \{6,8,12\}$, $(q+1,g)$-Moore graphs are $(q+1)$-regular graphs with girth $g$ which are the incidence graphs of a symmetric generalized $g/2$-gons of order $q$. In this paper we study the packing chromatic number of a $(q+1,g)$-Moore graph $G$. For $g=6$ we present the exact value of $χ_ρ(G)$. For $g=8$, we determine $χ_ρ(G)$ in terms of the intersection of certain structures in generalized quadrangles. For $g=12$, we present lower and upper bounds for this invariant when $q\ge 9$ an odd prime power.

math.CO↗

arXiv2018

The diachromatic number of digraphs

We consider the extension to directed graphs of the concept of achromatic number in terms of acyclic vertex colorings. The achromatic number have been intensely studied since it was introduced by Harary, Hedetniemi and Prins in 1967. The dichromatic number is a generalization of the chromatic number for digraphs defined by Neumann-Lara in 1982. A coloring of a digraph is an acyclic coloring if each subdigraph induced by each chromatic class is acyclic, and a coloring is complete if for any pair of chromatic classes $x,y$, there is an arc from $x$ to $y$ and an arc from $y$ to $x$. The dichromatic and diachromatic numbers are, respectively, the smallest and the largest number of colors in a complete acyclic coloring. We give some general results for the diachromatic number and study it for tournaments. We also show that the interpolation property for complete acyclic colorings does hold and establish Nordhaus-Gaddum relations.

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