Vertex-Magic Total Labeling on G-sun Graphs

Graph labeling is an immense area of research in mathematics, specifically graph theory. There are many types of graph labelings such as harmonious, magic, and lucky labelings. This paper will focus on magic labelings. Graph theorists are particularly interested in magic labelings because of a simple problem regarding tree graphs introduced in the 1990’s. The problem is still unsolved after almost thirty years. Researchers have studied magic labelings on other graphs in addition to tree graphs. In this paper we will consider vertex-magic labelings on G-sun graphs. We will give vertex-magic total labelings for ladder sun graphs and complete bipartite sun graphs. We will also show when there is no vertex-magic total labeling for other types of G-sun graphs

Some results concerning the valences of (super) edge-magic graphs

A graph $G$ is called edge-magic if there exists a bijective function $f:V\left(G\right) \cup E\left(G\right)\rightarrow \left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert +\left\vert E\left( G\right) \right\vert \right\}$ such that $f\left(u\right) + f\left(v\right) + f\left(uv\right)$ is a constant (called the valence of $f$) for each $uv\in E\left( G\right)$. If $f\left(V \left(G\right)\right) =\left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert \right\}$, then $G$ is called a super edge-magic graph. A stronger version of edge-magic and super edge-magic graphs appeared when the concepts of perfect edge-magic and perfect super edge-magic graphs were introduced. The super edge-magic deficiency $\mu_{s}\left(G\right)$ of a graph $G$ is defined to be either the smallest nonnegative integer $n$ with the property that $G \cup nK_{1}$ is super edge-magic or $+ \infty$ if there exists no such integer $n$. On the other hand, the edge-magic deficiency $\mu\left(G\right)$ of a graph $G$ is the smallest nonnegative integer $n$ for which $G\cup nK_{1}$ is edge-magic, being $\mu\left(G\right)$ always finite. In this paper, the concepts of (super) edge-magic deficiency are generalized using the concepts of perfect (super) edge-magic graphs. This naturally leads to the study of the valences of edge-magic and super edge-magic labelings. We present some general results in this direction and study the perfect (super) edge-magic deficiency of the star $K_{1,n}$

D4-Magic Graphs

Consider the set X = {1, 2, 3, 4} with 4 elements. A permutation of X is a function from X to itself that is both one one and on to. The permutations of X with the composition of functions as a binary operation is a nonabelian group, called the symmetric group S 4 . Now consider the collection of all permutations corresponding to the ways that two copies of a square with vertices 1, 2, 3 and 4 can be placed one covering the other with vertices on the top of vertices. This collection form a nonabelian subgroup of S 4 , called the dihedral group D 4 . In this paper, we introduce A-magic labelings of graphs, where A is a finite nonabelian group and investigate graphs that are D 4 -magic. This did not attract much attention in the literature

PEWARNAAN PELANGI ANTIAJAIB PADA GRAF PRISMA, GRAF ULAR SEGITIGA, DAN GRAF ULAR SEGITIGA GANDA

Teori graf adalah cabang matematika yang berhubungan dengan studi graf, yang merupakan struktur matematika yang mewakili hubungan antar objek. Teori graf pada konektivitas sudah banyak memberikan hasil yang kuat dan elegan, salah satunya adalah pewarnaan pelangi anti-ajaib. Pewarnaan pelangi anti-ajaib (rainbow anti-magic coloring) adalah konsep dalam teori graf yang berfokus pada pewarnaan sisi graf. Di dalam tulisan ini akan di bahas beberapa teorema baru beserta pembuktian koneksi pelangi (rainbow connection) dan pewarnaan pelangi anti-ajaib (rainbow anti-magic coloring) pada graf prisma, graf ular segitiga, dan graf ular segitiga ganda. Dalam memecahkan masalah, penelitian ini akan menggunakan metode deduktif yaitu metode yang berlaku dalam logika matematika dengan menggunakan aksioma atau teorema yang telah ada dan terbukti untuk memecahkan masalah. Hasil dari penelitian ini adalah 5 teorema yang terdiri dari : teorema pewarnaan pelangi anti-ajaib pada graf prisma yang terbagi dalam 8 kasus, secara umum ⌈n/2⌉ + m − 1 ≤ rc_A(Pr_{n,m}) ≤ ⌈n/2⌉ + 2m − 1, teorema koneksi pelangi pada graf ular segitiga yaitu rc(S(T_n)) = n − 1, teorema pewarnaan pelangi anti-ajaib pada graf ular segitiga yaitu rc_A(S(T_n)) = n + 1, teorema koneksi pelangi pada graf ular segitiga ganda yaitu rc(D(T_n)) = n, dan teorema pewarnaan pelangi anti-ajaib pada graf ular segitiga ganda yang terbagi dalam 2 kasus yaitu rc_A(D(T_2)) = 4 dan rc_A(D(T_n)) = n + 3, untuk n ≥ 3. ***** Graph theory is a branch of mathematics that deals with the study of graphs, which are mathematical structures that represent relationships between objects. Graph theory on connectivity has provided many powerful and elegant results, one of which is anti-magic rainbow coloring. Rainbow anti-magic coloring is a concept in theory a graph that focuses on coloring the edges of the graph. In this paper, several new theorems will be discussed along with the proof of rainbow connections and anti-magic rainbow coloring on prism graphs, triangular snake graphs, and double triangular snake graphs. In solving problems, this research will use the deductive method, which is a method that applies in mathematical logic by using existing and proven axioms or theorems to solve problems. The results of this research are 5 theorems consisting of: rainbow anti-magic coloring theorem on prism graphs which is divided into 8 cases, in general is ⌈n/2⌉+m−1 ≤ rc_A(Pr_{n,m}) ≤ ⌈n/2⌉+ 2m−1, rainbow connection theorem on triangular snake graphs that is rc(S(T_n)) = n − 1, rainbow anti-magic coloring theorem on triangular snake graphs that is rc_A(S(T_n)) = n+1, rainbow connection theorem on double triangular snake graphs that is rc(D(T_n)) = n, and the rainbow anti-magic coloring theorem on double triangular snake graph which is divided into 2 cases that is rc_A(D(T_2)) = 4 and rc_A(D(T_n)) = n + 3, for n ≥ 3

The spectra of VMT labelings of graphs

Tato práce se zabývá hledáním konstrukcí ohodnocení kompletních bipartitních grafů s vrcholově magickým totálním ohodnocením. Konkrétně takových ohodnocení, jejichž magická konstanta je na okraji spektra a neexistují pro ně doposud žádné konstrukce. Práce je rozdělena do několika kapitol, ve kterých si nejdříve představíme základní pojmy a počínaje pátou kapitolou si ukážeme několik přístupů, které je možno zvolit pro sestrojení ohodnocení grafů s magickou konstantou z hledaného spektra. Bohužel však nebudeme moci zaručit existenci VMT ohodnocení grafu $K_{n,n}$ pro libovolný řád $n$ a libovolnou magickou konstantou ze spektra.The thesis is about finding new constructions for complete bipartite graphs with vertex magic total labeling. Specifically labeling which has magic constant on the edge of spectrum and there are no known construction for them yet. Thesis is divided to several chapters. We will firstly define basic concept and ideas and starting with fifth chapter will be shown several approaches, which can be used for construnction of labeled graphs with magic constant from wanted spectrum. Unfortunatelly we can not guarantee the existence of VMT labeling of graph $K_{n,n}$ for any order $n$ with any magic constant from the spectrum.470 - Katedra aplikované matematikyvelmi dobř

Recent studies on the super edge-magic deficiency of graphs

A graph $G$ is called edge-magic if there exists a bijective function $f:V\left(G\right) \cup E\left(G\right)\rightarrow \left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert +\left\vert E\left( G\right) \right\vert \right\}$ such that $f\left(u\right) + f\left(v\right) + f\left(uv\right)$ is a constant for each $uv\in E\left( G\right)$. Also, $G$ is said to be super edge-magic if $f\left(V \left(G\right)\right) =\left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert \right\}$. Furthermore, the super edge-magic deficiency $\mu_{s}\left(G\right)$ of a graph $G$ is defined to be either the smallest nonnegative integer $n$ with the property that $G \cup nK_{1}$ is super edge-magic or $+ \infty$ if there exists no such integer $n$. In this paper, we introduce the parameter $l\left(n\right)$ as the minimum size of a graph $G$ of order $n$ for which all graphs of order $n$ and size at least $l\left(n\right)$ have $\mu_{s} \left( G \right)=+\infty$, and provide lower and upper bounds for $l\left(G\right)$. Imran, Baig, and Fe\u{n}ov\u{c}\'{i}kov\'{a} established that for integers $n$ with $n\equiv 0\pmod{4}$, $\mu_{s}\left(D_{n}\right) \leq 3n/2-1$, where $D_{n}$ is the cartesian product of the cycle $C_{n}$ of order $n$ and the complete graph $K_{2}$ of order $2$. We improve this bound by showing that $\mu_{s}\left(D_{n}\right) \leq n+1$ when $n \geq 4$ is even. Enomoto, Llad\'{o}, Nakamigawa, and Ringel posed the conjecture that every nontrivial tree is super edge-magic. We propose a new approach to attak this conjecture. This approach may also help to resolve another labeling conjecture on trees by Graham and Sloane

Utilization of 1-VMV labelings of graphs

Import 03/08/2012Tato práce je zaměřena na využití grafů s takzvaným 1-VMV ohodnocením, pro které se v poslední době vžil alternativní název distance magic ohodnocení. Práce shrnuje známé výsledky v oblasti distance magic ohodnocení grafů, ovšem hlavním přínosem je popis induktivní konstrukce distance magic grafů, které lze použít při plánování neúplných sportovních turnajů. Navíc oproti zadání bakalářské práce je zde část zaměřená na možnost určité přímé či nepřímé dominance v těchto turnajích, která zajistí každému účastníkovi turnaje (hráči či týmu) stejnou šanci stát se tzv. králem turnaje. Tato problematika nebyla zatím v žádné literatuře řešena.This thesis is focused on the use of graphs with a 1-VMV labeling, which is recently called distance magic labeling. Thesis sums up known results in the area of distance magic graphs which can be used for scheduling incomplete sports tournaments. In addition to the assigment of thesis there is a part focused on possibility of direct or indirect domination in these tournaments to ensure every member in tournament (player or team) can become a so--called king of tournament. This issue has not been addressed in any literature before.470 - Katedra aplikované matematikyvýborn