Drawing a Graph in a Hypercube

A $d$-dimensional hypercube drawing of a graph represents the vertices by distinct points in $\{0,1\}^d$, such that the line-segments representing the edges do not cross. We study lower and upper bounds on the minimum number of dimensions in hypercube drawing of a given graph. This parameter turns out to be related to Sidon sets and antimagic injections.Comment: Submitte

On \u3cem\u3ek\u3c/em\u3e-minimum and \u3cem\u3em\u3c/em\u3e-minimum Edge-Magic Injections of Graphs

An edge-magic total labelling (EMTL) of a graph G with n vertices and e edges is an injection λ:V(G) ∪ E(G)→[n+e], where, for every edge uv ∈ E(G), we have wtλ(uv)=kλ, the magic sum of λ. An edge-magic injection (EMI) μ of G is an injection μ : V(G) ∪ E(G) → N with magic sum kμ and largest label mμ. For a graph G we define and study the two parameters κ(G): the smallest kμ amongst all EMI’s μ of G, and m(G): the smallest mμ amongst all EMI’s μ of G. We find κ(G) for G ∈ G for many classes of graphs G. We present algorithms which compute the parameters κ(G) and m(G). These algorithms use a G-sequence: a sequence of integers on the vertices of G whose sum on edges is distinct. We find these parameters for all G with up to 7 vertices. We introduce the concept of a double-witness: an EMI μ of G for which both kμ=κ(G) and mμ=m(G) ; and present an algorithm to find all double-witnesses for G. The deficiency of G, def(G), is m(G)−n−e. Two new graphs on 6 vertices with def(G)=1 are presented. A previously studied parameter of G is κEMTL(G), the magic strength of G: the smallest kλ amongst all EMTL’s λ of G. We relate κ(G) to κEMTL(G) for various G, and find a class of graphs B for which κEMTL(G)−κ(G) is a constant multiple of n−4 for G ∈B. We specialise to G=Kn, and find both κ(Kn) and m(Kn) for all n≤11. We relate κ(Kn) and m(Kn) to known functions of n, and give lower bounds for κ(Kn) and m(Kn)

Minimum-Weight Edge Discriminator in Hypergraphs

In this paper we introduce the concept of minimum-weight edge-discriminators in hypergraphs, and study its various properties. For a hypergraph $\mathcal H=(\mathcal V, \mathcal E)$, a function $\lambda: \mathcal V\rightarrow \mathbb Z^{+}\cup\{0\}$ is said to be an {\it edge-discriminator} on $\mathcal H$ if $\sum_{v\in E_i}{\lambda(v)}>0$, for all hyperedges $E_i\in \mathcal E$, and $\sum_{v\in E_i}{\lambda(v)}\ne \sum_{v\in E_j}{\lambda(v)}$, for every two distinct hyperedges $E_i, E_j \in \mathcal E$. An {\it optimal edge-discriminator} on $\mathcal H$, to be denoted by $\lambda_\mathcal H$, is an edge-discriminator on $\mathcal H$ satisfying $\sum_{v\in \mathcal V}\lambda_\mathcal H (v)=\min_\lambda\sum_{v\in \mathcal V}{\lambda(v)}$, where the minimum is taken over all edge-discriminators on $\mathcal H$. We prove that any hypergraph $\mathcal H=(\mathcal V, \mathcal E)$, with $|\mathcal E|=n$, satisfies $\sum_{v\in \mathcal V} \lambda_\mathcal H(v)\leq n(n+1)/2$, and equality holds if and only if the elements of $\mathcal E$ are mutually disjoint. For $r$-uniform hypergraphs $\mathcal H=(\mathcal V, \mathcal E)$, it follows from results on Sidon sequences that $\sum_{v\in \mathcal V}\lambda_{\mathcal H}(v)\leq |\mathcal V|^{r+1}+o(|\mathcal V|^{r+1})$, and the bound is attained up to a constant factor by the complete $r$-uniform hypergraph. Next, we construct optimal edge-discriminators for some special hypergraphs, which include paths, cycles, and complete $r$-partite hypergraphs. Finally, we show that no optimal edge-discriminator on any hypergraph $\mathcal H=(\mathcal V, \mathcal E)$, with $|\mathcal E|=n (\geq 3)$, satisfies $\sum_{v\in \mathcal V} \lambda_\mathcal H (v)=n(n+1)/2-1$, which, in turn, raises many other interesting combinatorial questions.Comment: 22 pages, 5 figure

Kekuatan tak beraturan sisi total pada graf hasil gabungan graf lintasan dengan beberapa kelas graf

Penelitian ini bertujuan untuk menentukan nilai kekuatan tak beraturan sisi total pada graf hasil gabungan graf lingkaran dengan graf lintasan dan graf hasil gabungan dua graf lingkaran masing-masing untuk n sama dengan 3. Pelabelan tak beraturan sisi total pada graf, dengan himpunan titik tak kosong V dan himpunan sisi E suatu fungsi, sehingga bobot setiap sisinya berbeda. Nilai k terkecil pada pelabelan tak beraturan sisi total disebut kekuatan tak beraturan sisi total dari G yang dinotasikan dengan tes G. Selanjutnya, bobot sebuah sisi uv dengan fungsi pelabelan. Berdasarkan pembahasan, dapat disimpulkan bahwa nilai kekuatan tak beraturan sisi total pada graf hasil gabungan graf lingkaran dengan graf lintasan dan graf hasil gabungan dua graf lingkaran yang berturut-turut mempunyai nilai n sama dengan 3

Minimum-Weight Edge Discriminators in Hypergraphs

In this paper we introduce the notion of minimum-weight edge-discriminators in hypergraphs, and study their various properties. For a hypergraph H = (V , E), a function λ : V → Z+∪{0} is said to be an edge-discriminator on H if ∑v∈Eiλ(v)\u3e0, for all hyperedges Ei ∈ E and ∑v∈Eiλ(v) ≠ ∑v∈Ejλ(v), for every two distinct hyperedges Ei,Ej, ∈ E. An optimal edge-discriminator on H, to be denoted by λH, is an edge-discriminator on H satisfying ∑v∈VλH(v) = minλ ∑v∈Vλ(v), where the minimum is taken over all edge-discriminators on H. We prove that any hypergraph H = (V , E), with |E| = m, satisfies ∑v∈VλH(v) ≤ m(m+1)/2, and the equality holds if and only if the elements of E are mutually disjoint. For r-uniform hypergraphs H = (V,E), it follows from earlier results on Sidon sequences that ∑v∈VλH(v) ≤ |V|r+1+o(|V|r+1), and the bound is attained up to a constant factor by the complete r-uniform hypergraph. Finally, we show that no optimal edge-discriminator on any hypergraph H = (V,E), with |E| = m (≥3), satisfies ∑v∈VλH(v) = m(m+1)/2−1. This shows that all integer values between m and m(m+1)/2 cannot be the weight of an optimal edge-discriminator of a hypergraph, and this raises many other interesting combinatorial questions