Temat: (P,Q)-total coloring - Katalog OPAC zbiorów

Skocz do pozycji: 1.

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

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Skocz do pozycji: 2.

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

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