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    Wyświetlanie 1-10 z 10
    Tytuł:
    Edge colorings and total colorings of integer distance graphs
    Autorzy:
    Kemnitz, Arnfried
    Marangio, Massimiliano
    Powiązania:
    https://bibliotekanauki.pl/articles/743555.pdf
    Data publikacji:
    2002
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    integer distance graph
    chromatic number
    choice number
    chromatic index
    choice index
    total chromatic number
    total choice number
    Opis:
    An integer distance graph is a graph G(D) with the set Z of integers as vertex set and two vertices u,v ∈ Z are adjacent if and only if |u-v| ∈ D where the distance set D is a subset of the positive integers N. In this note we determine the chromatic index, the choice index, the total chromatic number and the total choice number of all integer distance graphs, and the choice number of special integer distance graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2002, 22, 1; 149-158
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Graphs with rainbow connection number two
    Autorzy:
    Kemnitz, Arnfried
    Schiermeyer, Ingo
    Powiązania:
    https://bibliotekanauki.pl/articles/743883.pdf
    Data publikacji:
    2011
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    edge colouring
    rainbow colouring
    rainbow connection
    Opis:
    An edge-coloured graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colours. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colours that are needed in order to make G rainbow connected. In this paper we prove that rc(G) = 2 for every connected graph G of order n and size m, where $\binom{n-1}{2} + 1 ≤ m ≤ \binom{n}{2} - 1$. We also characterize graphs with rainbow connection number two and large clique number.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2011, 31, 2; 313-320
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On the ρ-Edge Stability Number of Graphs
    Autorzy:
    Kemnitz, Arnfried
    Marangio, Massimiliano
    Powiązania:
    https://bibliotekanauki.pl/articles/32361738.pdf
    Data publikacji:
    2022-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    edge stability number
    line stability
    invariant
    chromatic edge stability index
    chromatic index
    edge coloring
    Opis:
    For an arbitrary invariant $ \rho(G) $ of a graph $G$ the $ \rho $-edge stability number $ es_\rho (G) $ is the minimum number of edges of $G$ whose removal results in a graph $ H \subseteq G $ with $ \rho (H) \ne \rho (G) $ or with $ E(H) = \emptyset $. In the first part of this paper we give some general lower and upper bounds for the $ \rho $-edge stability number. In the second part we study the $ \chi^' $-edge stability number of graphs, where $ \chi^' = \chi^' (G) $ is the chromatic index of $G$. We prove some general results for the so-called chromatic edge stability index $ es_{ \chi^′ } (G) $ and determine $ es_{ \chi^′ } (G) $ exactly for specific classes of graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2022, 42, 1; 249-262
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Fractional (\( \mathcal{P} , \mathcal{Q} \))-Total List Colorings of Graphs
    Autorzy:
    Kemnitz, Arnfried
    Mihók, Peter
    Voigt, Margit
    Powiązania:
    https://bibliotekanauki.pl/articles/30146708.pdf
    Data publikacji:
    2013-03-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    graph property
    total coloring
    (P,Q)-total coloring
    fractional coloring
    fractional (P,Q)-total chromatic number
    circular coloring
    circular (P,Q)-total chromatic number
    list coloring
    (P,Q)-total (a
    b)-list colorings
    Opis:
    Let $ r, s \in \mathbb{N}$, $r \ge s$, and \( \mathcal{P} \) and \( \mathcal{Q} \) be two additive and hereditary graph properties. A \( (P,Q) \)-total $(r, s)$-coloring of a graph $G = (V,E)$ is a coloring of the vertices and edges of $G$ by $s$-element subsets of $ \mathbb{Z}_r$ such that for each color $i$, $0 \le i \le r − 1$, the vertices colored by subsets containing $i$ induce a subgraph of $G$ with property \( \mathcal{P} \), the edges colored by subsets containing $i$ induce a subgraph of $G$ with property \( \mathcal{Q} \), and color sets of incident vertices and edges are disjoint. The fractional \( (\mathcal{P}, \mathcal{Q})\)-total chromatic number $ \chi_{f,P,Q}^{''}(G)$ of $G$ is defined as the infimum of all ratios $r//s$ such that $G$ has a \( ( \mathcal{P}, \mathcal{Q})\)-total $(r, s)$-coloring. A \( ( \mathcal{P}, \mathcal{Q} \)-total independent set $ T = V_T \cup E_T \subseteq V \cup E$ is the union of a set $V_T$ of vertices and a set $E_T$ of edges of $G$ such that for the graphs induced by the sets $V_T$ and $E_T$ it holds that \( G[ V_T ] \in \mathcal{ P } \), \( G[ E_T ] \in \mathcal{Q} \), and $ G[ V_T ] $ and $ G[ E_T ] $ are disjoint. Let \( T_{ \mathcal{P} , \mathcal{Q} } \) be the set of all \( (\mathcal{P} ,\mathcal{Q})\)-total independent sets of $G$. Let $L(x)$ be a set of admissible colors for every element $ x \in V \cup E $. The graph $G$ is called \( (\mathcal{P} , \mathcal{Q}) \)-total $(a, b)$-list colorable if for each list assignment $L$ with $|L(x)| = a$ for all $x \in V \cup E$ it is possible to choose a subset $ C(x) \subseteq L(x)$ with $|C(x)| = b$ for all $ x \in V \cup E$ such that the set $ T_i $ which is defined by $ T_i = {x \in V \cup E : i \in C(x) } $ belongs to \( T_{ \mathcal{P},\mathcal{Q}}\) for every color $i$. The \( (\mathcal{P}, \mathcal{Q})\)- choice ratio \( \text{chr}_{\mathcal{P},\mathcal{Q}}(G)\) of $G$ is defined as the infimum of all ratios $a//b$ such that $G$ is \( (\mathcal{P},\mathcal{Q})\)-total $(a, b)$-list colorable. We give a direct proof of \( \chi_{ f,\mathcal{P},\mathcal{Q} }^{ \prime \prime } (G) = \text{chr}_{ \mathcal{P} ,\mathcal{Q} }(G)\) for all simple graphs $G$ and we present for some properties \( \mathcal{P} \) and \( \mathcal{Q} \) new bounds for the \( (\mathcal{P}, \mathcal{Q})\)-total chromatic number and for the \((\mathcal{P},\mathcal{Q})\)-choice ratio of a graph $G$.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2013, 33, 1; 167-179
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Sum List Edge Colorings of Graphs
    Autorzy:
    Kemnitz, Arnfried
    Marangio, Massimiliano
    Voigt, Margit
    Powiązania:
    https://bibliotekanauki.pl/articles/31340809.pdf
    Data publikacji:
    2016-08-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    sum list edge coloring
    sum choice index
    sum list coloring
    sum choice number
    choice function
    line graph
    Opis:
    Let $ G = (V,E) $ be a simple graph and for every edge $ \mathcal{e} \in E $ let $ L(e) $ be a set (list) of available colors. The graph $ G $ is called $L$-edge colorable if there is a proper edge coloring $ c $ of $ G $ with $ c(\mathcal{e} ) \in L( \mathcal{e} ) $ for all $ \mathcal{e} \in E $. A function $ f : E \rightarrow \mathbb{N} $ is called an edge choice function of $G$ and $G$ is said to be $f$-edge choosable if $G$ is $L$-edge colorable for every list assignment $L$ with $ |L( \mathcal{e} )| = f( \mathcal{e} ) $ for all $ \mathcal{e} \in E $. Set $ \text{size}(f) = \Sigma_{ \mathcal{e} \in E } f(e) $ and define the sum choice index $ \chi_{sc}^' (G) $ as the minimum of $ \text{size} (f) $ over all edge choice functions $f$ of $G$. There exists a greedy coloring of the edges of $G$ which leads to the upper bound $ \chi_{sc}^′ (G) \le 1/2 \Sigma_{ v \in V } d(v)^2 $. A graph is called sec-greedy if its sum choice index equals this upper bound. We present some general results on the sum choice index of graphs including a lower bound and we determine this index for several classes of graphs. Moreover, we present classes of sec-greedy graphs as well as all such graphs of order at most 5.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2016, 36, 3; 709-722
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Generalized Sum List Colorings of Graphs
    Autorzy:
    Kemnitz, Arnfried
    Marangio, Massimiliano
    Voigt, Margit
    Powiązania:
    https://bibliotekanauki.pl/articles/31343297.pdf
    Data publikacji:
    2019-08-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    sum list coloring
    sum choice number
    generalized sum list coloring
    additive hereditary graph property
    Opis:
    A (graph) property \( \mathcal{P} \) is a class of simple finite graphs closed under isomorphisms. In this paper we consider generalizations of sum list colorings of graphs with respect to properties \( \mathcal{P} \). If to each vertex $v$ of a graph $G$ a list $L(v)$ of colors is assigned, then in an \( (L, \mathcal{P} ) \)-coloring of $G$ every vertex obtains a color from its list and the subgraphs of $G$ induced by vertices of the same color are always in \( \mathcal{P} \). The \( \mathcal{P} \)-sum choice number \( X_{sc}^\mathcal{P} (G) \) of $G$ is the minimum of the sum of all list sizes such that, for any assignment $L$ of lists of colors with the given sizes, there is always an \( (L, \mathcal{P} ) \)-coloring of $G$. We state some basic results on monotonicity, give upper bounds on the \( \mathcal{P} \)-sum choice number of arbitrary graphs for several properties, and determine the \( \mathcal{P} \)-sum choice number of specific classes of graphs, namely, of all complete graphs, stars, paths, cycles, and all graphs of order at most 4.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2019, 39, 3; 689-703
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Improved Sufficient Conditions for Hamiltonian Properties
    Autorzy:
    Bode, Jens-P.
    Fricke, Anika
    Kemnitz, Arnfried
    Powiązania:
    https://bibliotekanauki.pl/articles/31339476.pdf
    Data publikacji:
    2015-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    Hamiltonian
    traceable
    Hamiltonian-connected
    Opis:
    In 1980 Bondy [2] proved that a (k+s)-connected graph of order n ≥ 3 is traceable (s = −1) or Hamiltonian (s = 0) or Hamiltonian-connected (s = 1) if the degree sum of every set of k+1 pairwise nonadjacent vertices is at least ((k+1)(n+s−1)+1)/2. It is shown in [1] that one can allow exceptional (k+1)-sets violating this condition and still implying the considered Hamiltonian property. In this note we generalize this result for s = −1 and s = 0 and graphs that fulfill a certain connectivity condition.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2015, 35, 2; 329-334
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Rainbow Connection In Sparse Graphs
    Autorzy:
    Kemnitz, Arnfried
    Przybyło, Jakub
    Schiermeyer, Ingo
    Woźniak, Mariusz
    Powiązania:
    https://bibliotekanauki.pl/articles/30146690.pdf
    Data publikacji:
    2013-03-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    rainbow-connected graph
    rainbow colouring
    rainbow connection number
    Opis:
    An edge-coloured connected graph $G = (V,E)$ is called rainbow-connected if each pair of distinct vertices of $G$ is connected by a path whose edges have distinct colours. The rainbow connection number of $G$, denoted by $ \text{rc}(G)$, is the minimum number of colours such that $G$ is rainbow-connected. In this paper we prove that $ \text{rc}(G) \le k $ if $ |V (G)| = n $ and \( |E(G)| \ge \binom{n-k+1}{2} + k -1 \) for all integers $n$ and $k$ with $n − 6 \le k \le n − 3 $. We also show that this bound is tight.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2013, 33, 1; 181-192
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Generalized total colorings of graphs
    Autorzy:
    Borowiecki, Mieczysław
    Kemnitz, Arnfried
    Marangio, Massimiliano
    Mihók, Peter
    Powiązania:
    https://bibliotekanauki.pl/articles/743865.pdf
    Data publikacji:
    2011
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    hereditary properties
    generalized total colorings
    paths
    cycles
    complete graphs
    Opis:
    An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let P and Q be additive hereditary properties of graphs. A (P,Q)-total coloring of a simple graph G is a coloring of the vertices V(G) and edges E(G) of G such that for each color i the vertices colored by i induce a subgraph of property P, the edges colored by i induce a subgraph of property Q and incident vertices and edges obtain different colors. In this paper we present some general basic results on (P,Q)-total colorings. We determine the (P,Q)-total chromatic number of paths and cycles and, for specific properties, of complete graphs. Moreover, we prove a compactness theorem for (P,Q)-total colorings.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2011, 31, 2; 209-222
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Generalized Fractional and Circular Total Colorings of Graphs
    Autorzy:
    Kemnitz, Arnfried
    Marangio, Massimiliano
    Mihók, Peter
    Oravcová, Janka
    Soták, Roman
    Powiązania:
    https://bibliotekanauki.pl/articles/31339338.pdf
    Data publikacji:
    2015-08-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    graph property
    (P,Q)-total coloring
    circular coloring
    fractional coloring
    fractional (P,Q)-total chromatic number
    circular (P,Q)- total chromatic number
    Opis:
    Let \( \mathcal{P} \) and \( \mathcal{Q} \) be additive and hereditary graph properties, $ r, s \in \mathbb{N}$, $ r \ge s $, and $ [\mathbb{Z}_r]^s $ be the set of all s-element subsets of $\mathbb{Z}_r $. An ($r$, $s$)-fractional (\( \mathcal{P} \),\( \mathcal{Q} \))-total coloring of $G$ is an assignment $ h : V (G) \cup E(G) \rightarrow [\mathbb{Z}_r]^s $ such that for each $ i \in \mathbb{Z}_r $ the following holds: the vertices of $G$ whose color sets contain color $i$ induce a subgraph of $G$ with property \( \mathcal{P} \), edges with color sets containing color $i$ induce a subgraph of $G$ with property \( \mathcal{Q} \), and the color sets of incident vertices and edges are disjoint. If each vertex and edge of $G$ is colored with a set of $s$ consecutive elements of $ \mathbb{Z}_r $ we obtain an ($r$, $s$)-circular (\( \mathcal{P} \),\( \mathcal{Q} \))-total coloring of $G$. In this paper we present basic results on ($r$, $s$)-fractional/circular (\( \mathcal{P} \),\( \mathcal{Q} \))-total colorings. We introduce the fractional and circular (\( \mathcal{P} \),\( \mathcal{Q}\))-total chromatic number of a graph and we determine this number for complete graphs and some classes of additive and hereditary properties.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2015, 35, 3; 517-532
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-10 z 10

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