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On local antimagic total labeling of complete graphs amalgamation

Lau Gee-Choon, Shiu Wai-Chee

Opuscula Mathematica

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2023

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Vol. 43, no. 3

429--453

EN

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) ̸= g+(v), where g+(u) = ∑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 wf (u) ̸= wf (v), where wf (u) = f(u) + ∑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, wf (u)). The local antimagic (total) chromatic number, denoted χla(G) (respectively χlat(G)), is the minimum number of induced colors taken over local antimagic (total) labeling of G. In this paper, we determined χlat(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 K1 and amalgamation of complete graphs under various conditions. An application of local antimagic total chromatic number is also given.

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The strong 3-rainbow index of some certain graphs and its amalgamation

Awanis Zata Yumni, Salman A.N.M.

Opuscula Mathematica

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2022

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Vol. 42, no. 4

527--547

EN

We introduce a strong k-rainbow index of graphs as modification of well-known k-rainbow index of graphs. A tree in an edge-colored connected graph G, where adjacent edge may be colored the same, is a rainbow tree if all of its edges have distinct colors. Let k be an integer with 2 ≤ k ≤ n. The strong k-rainbow index of G, denoted by srxk(G), is the minimum number of colors needed in an edge-coloring of G so that every k vertices of G is connected by a rainbow tree with minimum size. We focus on k = 3. We determine the strong 3-rainbow index of some certain graphs. We also provide a sharp upper bound for the strong 3-rainbow index of amalgamation of graphs. Additionally, we determine the exact values of the strong 3-rainbow index of amalgamation of some graphs.

3

Charakterystyka złota Hg - nośnego z wybranych obszarów Dolnego Śląska

Kania M., Muszer A.

Górnictwo Odkrywkowe

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2017

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R. 58, nr 5

11--21

PL

Z aluwiów korytowych Przedgórza Kaczawskiego i obszarów sąsiednich pozyskano ziarna złota rtęciowego z domieszką Hg do 6,95 % wag. Obecność rtęci najczęściej związana jest z partiami złota o wyższych zawartościach Ag. Ziarna podzielić można na podstawie różnic teksturalnych i składu chemicznego na dwie grupy. Pierwszą stanowią złociny o teksturze gąbczastej rozwiniętej w całej objętości ziaren, o zawartości Hg nie przekraczającej < 2 % wag. Złoto tego typu stanowi produkt procesu amalgamacji i jako efekt działalności antropogenicznej może występować we wszystkich znanych wystąpieniach złota rozsypiskowego. Drugą stanowią złociny o bimodalnej budowie, złożone ze stopu Au-Ag o składzie fazy metastabilnej lub elektrum i czystego złota rodzimego. Faza złota rodzimego nie zawiera rtęci. Faza srebrowa ma zwartą teksturę, a koncentracja Hg sięga kilku % wag. Prawdopodobnym obszarem alimentacyjnym tego typu złota są wychodnie stref kontaktu czerwonego spągowca - cechsztynu.

EN

Particles of mercurial gold with Hg content of max. 6,95 wt % have been found in the riverbeds of the Kaczawskie Foothils and adjacent areas. The presence of mercury is predominantly connected with the parts of gold with higher concentration of silver. Due to structural and chemical differences two types of mercurial gold can be distinguished. The first one comprises gold particles with spongy structure developed in the whole volume of grains, in which the Hg concentration doesn’t exceed 2 wt %. The gold of this type represents the product of the amalgamation process and, as an effect of anthropogenic activity, can occur in the whole known placer gold deposits. The second one comprises gold particles of bimodal structure, made of Au-Ag alloy of unstable or electrum phase and phase of pure gold. The phase of pure gold doesn’t contain Hg. The phase of Ag-bearing gold has a dense structure with a Hg concentration of a few wt %. The outcrops of the contact zones between Rotliegend and Zechstein are the most probable sources of this type of gold.