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    Wyświetlanie 1-8 z 8
    Tytuł:
    On local antimagic total labeling of complete graphs amalgamation
    Autorzy:
    Lau, Gee-Choon
    Shiu, Wai-Chee
    Powiązania:
    https://bibliotekanauki.pl/articles/29519348.pdf
    Data publikacji:
    2023
    Wydawca:
    Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
    Tematy:
    local antimagic (total) chromatic number
    amalgamation
    complete graph
    Opis:
    Let G = (V,E) be a connected simple graph of order p and size q. A graph G is called local antimagic (total) if G admits a local antimagic (total) labeling. A bijection g : E → {1, 2, . . . , q} is called a local antimagic labeling of G if for any two adjacent vertices u and v, we have $ g^+ (u) \ne g^+ (v) $, where $ g^+ (u) = \Sigma_{e∈E(u)} \text{ } g(e) $, and E(u) is the set of edges incident to u. Similarly, a bijection f : V (G)∪E(G) → {1, 2, . . . , p+q} is called a local antimagic total labeling of G if for any two adjacent vertices u and v, we have $ w_f (u) \ne w_f (v) $, where $ w_f (u) = f(u) + \Sigma_{e∈E(u)} f(e) $. Thus, any local antimagic (total) labeling induces a proper vertex coloring of G if vertex v is assigned the color $g^+ (v) $ (respectively, $ w_f (u) $). The local antimagic (total) chromatic number, denoted $χ_\text{la } (G) $ (respectively $χ_\text{lat } (G) $ ), is the minimum number of induced colors taken over local antimagic (total) labeling of G. In this paper, we determined $ χ_\text{lat } (G) $ where G is the amalgamation of complete graphs. Consequently, we also obtained the local antimagic (total) chromatic number of the disjoint union of complete graphs, and the join of $ K_1 $ and amalgamation of complete graphs under various conditions. An application of local antimagic total chromatic number is also given.
    Źródło:
    Opuscula Mathematica; 2023, 43, 3; 429-453
    1232-9274
    2300-6919
    Pojawia się w:
    Opuscula Mathematica
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Chromatic Properties of the Pancake Graphs
    Autorzy:
    Konstantinova, Elena
    Powiązania:
    https://bibliotekanauki.pl/articles/31341647.pdf
    Data publikacji:
    2017-08-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    Pancake graph
    Cayley graphs
    symmetric group
    chromatic number
    total chromatic number
    Opis:
    Chromatic properties of the Pancake graphs Pn, n ⩾ 2, that are Cayley graphs on the symmetric group Symn generated by prefix-reversals are investigated in the paper. It is proved that for any n ⩾ 3 the total chromatic number of Pn is n, and it is shown that the chromatic index of Pn is n − 1. We present upper bounds on the chromatic number of the Pancake graphs Pn, which improve Brooks’ bound for n ⩾ 7 and Katlin’s bound for n ⩽ 28. Algorithms of a total n-coloring and a proper (n − 1)-coloring are given.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2017, 37, 3; 777-787
    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
    Tytuł:
    Every graph is local antimagic total and its applications
    Autorzy:
    Lau, Gee-Choon
    Schaffer, Karl
    Shiu, Wai-Chee
    Powiązania:
    https://bibliotekanauki.pl/articles/29519430.pdf
    Data publikacji:
    2023
    Wydawca:
    Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
    Tematy:
    local antimagic (total) chromatic number
    Cartesian product
    join product
    Opis:
    Let $ G = (V, E) $ be a simple graph of order $ p $ and size $ q $. A graph $ G $ is called local antimagic (total) if $ G $ admits a local antimagic (total) labeling. A bijection $ g : E → {1, 2, . . . , q} $ is called a local antimagic labeling of $ G $ if for any two adjacent vertices $ u $ and $ v $, we have $g^+(u) ≠ g^+(v) $, where $ g^+(u) = \Sigma_{e∈E(u)} g(e) $, and $ E(u) $ is the set of edges incident to $ u $. Similarly, a bijection $f : V (G)∪E(G) → {1, 2, . . . , p+q} $ is called a local antimagic total labeling of $ G $ if for any two adjacent vertices $ u $ and $ v $, we have $ w_f (u) ≠ w_f (v) $, where $ w_f (u) = f(u) + \Sigma_{e∈E(u)} f(e) $. Thus, any local antimagic (total) labeling induces a proper vertex coloring of $ G $ if vertex $ v $ is assigned the color $ g^+ (v) $ (respectively, $ w_f (u) $). The local antimagic (total) chromatic number, denoted $ χ_{la} (G) $ (respectively $ χ_{lat} (G)$ ), is the minimum number of induced colors taken over local antimagic (total) labeling of $ G $. We provide a short proof that every graph $ G $ is local antimagic total. The proof provides sharp upper bound to $ χ_{lat} (G) $. We then determined the exact $ χ_{lat} (G) $, where $ G $ is a complete bipartite graph, a path, or the Cartesian product of two cycles. Consequently, the $ χ_{la} (G ∨ K_1) $ is also obtained. Moreover, we determined the $ χ_{la} (G ∨ K_1) $ and hence the $χ_{lat} (G) $ for a class of 2-regular graphs $ G $ (possibly with a path). The work of this paper also provides many open problems on $ χ_{lat} (G) $. We also conjecture that each graph $ G $ of order at least 3 has $ χ_{lat} (G) ≤ χ_{la} (G) $.
    Źródło:
    Opuscula Mathematica; 2023, 43, 6; 841-864
    1232-9274
    2300-6919
    Pojawia się w:
    Opuscula Mathematica
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    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:
    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:
    Vertex-Distinguishing IE-Total Colorings of Complete Bipartite Graphs Km,N(m < n)
    Autorzy:
    Chen, Xiang’en
    Gao, Yuping
    Yao, Bing
    Powiązania:
    https://bibliotekanauki.pl/articles/30146641.pdf
    Data publikacji:
    2013-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    complete bipartite graphs
    IE-total coloring
    vertex-distinguishing IE-total coloring
    vertex-distinguishing IE-total chromatic number
    Opis:
    Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. Let C(u) be the set of colors of vertex u and edges incident to u under f. For an IE-total coloring f of G using k colors, if C(u) ≠ C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-coloring of G, or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χievt(G), and is called vertex-distinguishing IE-total chromatic number or the VDIET chromatic number of G for short. VDIET colorings of complete bipartite graphs Km,n(m < n) are discussed in this paper. Particularly, the VDIET chromatic numbers of Km,n(1 ≤ m ≤ 7, m < n) as well as complete graphs Kn are obtained.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2013, 33, 2; 289-306
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On the Total Graph of Mycielski Graphs, Central Graphs and Their Covering Numbers
    Autorzy:
    Patil, H.P.
    Pandiya Raj, R.
    Powiązania:
    https://bibliotekanauki.pl/articles/30146583.pdf
    Data publikacji:
    2013-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    total graph
    central graph
    middle graph
    Mycielski graph
    independence number
    covering number
    edge independence number
    edge covering number
    chromatic number
    achromatic number
    Opis:
    The technique of counting cliques in networks is a natural problem. In this paper, we develop certain results on counting of triangles for the total graph of the Mycielski graph or central graph of star as well as completegraph families. Moreover, we discuss the upper bounds for the number of triangles in the Mycielski and other well known transformations of graphs. Finally, it is shown that the achromatic number and edge-covering number of the transformations mentioned above are equated.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2013, 33, 2; 361-371
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-8 z 8

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