Sigma Partitioning: Complexity and Random Graphs

A $\textit{sigma partitioning}$ of a graph $G$ is a partition of the vertices into sets $P_1, \ldots, P_k$ such that for every two adjacent vertices $u$ and $v$ there is an index $i$ such that $u$ and $v$ have different numbers of neighbors in $P_i$. The $\textit{ sigma number}$ of a graph $G$, denoted by $\sigma(G)$, is the minimum number $k$ such that $G$ has a sigma partitioning $P_1, \ldots, P_k$. Also, a $\textit{ lucky labeling}$ of a graph $G$ is a function $\ell :V(G) \rightarrow \mathbb{N}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w)$ ($x \sim y$ means that $x$ and $y$ are adjacent). The $\textit{ lucky number}$ of $G$, denoted by $\eta(G)$, is the minimum number $k$ such that $G$ has a lucky labeling $\ell :V(G) \rightarrow \mathbb{N}_k$. It was conjectured in [Inform. Process. Lett., 112(4):109--112, 2012] that it is $\mathbf{NP}$-complete to decide whether $\eta(G)=2$ for a given 3-regular graph $G$. In this work, we prove this conjecture. Among other results, we give an upper bound of five for the sigma number of a uniformly random graph

On the Complexity of Digraph Colourings and Vertex Arboricity

It has been shown by Bokal et al. that deciding 2-colourability of digraphs is an NP-complete problem. This result was later on extended by Feder et al. to prove that deciding whether a digraph has a circular $p$-colouring is NP-complete for all rational $p>1$. In this paper, we consider the complexity of corresponding decision problems for related notions of fractional colourings for digraphs and graphs, including the star dichromatic number, the fractional dichromatic number and the circular vertex arboricity. We prove the following results: Deciding if the star dichromatic number of a digraph is at most $p$ is NP-complete for every rational $p>1$. Deciding if the fractional dichromatic number of a digraph is at most $p$ is NP-complete for every $p>1, p \neq 2$. Deciding if the circular vertex arboricity of a graph is at most $p$ is NP-complete for every rational $p>1$. To show these results, different techniques are required in each case. In order to prove the first result, we relate the star dichromatic number to a new notion of homomorphisms between digraphs, called circular homomorphisms, which might be of independent interest. We provide a classification of the computational complexities of the corresponding homomorphism colouring problems similar to the one derived by Feder et al. for acyclic homomorphisms.Comment: 21 pages, 1 figur

The sigma chromatic number of the Sierpinski gasket graphs and the Hanoi graphs

A vertex coloring c : V(G) → of a non-trivial connected graph G is called a sigma coloring if σ(u) ≠ σ(v) for any pair of adjacent vertices u and v. Here, σ(x) denotes the sum of the colors assigned to vertices adjacent to x. The sigma chromatic number of G, denoted by σ(G), is defined as the fewest number of colors needed to construct a sigma coloring of G. In this paper, we determine the sigma chromatic numbers of the Sierpiński gasket graphs and the Hanoi graphs. Moreover, we prove the uniqueness of the sigma coloring for Sierpiński gasket graphs

Sigma chromatic number of graph coronas involving complete graphs

Let c : V(G) → be a coloring of the vertices in a graph G. For a vertex u in G, the color sum of u, denoted by σ(u), is the sum of the colors of the neighbors of u. The coloring c is called a sigma coloring of G if σ(u) ≠ σ(v) whenever u and v are adjacent vertices in G. The minimum number of colors that can be used in a sigma coloring of G is called the sigma chromatic number of G and is denoted by σ(G). Given two simple, connected graphs G and H, the corona of G and H, denoted by G ⊙ H, is the graph obtained by taking one copy of G and |V(G)| copies of H and where the ith vertex of G is adjacent to every vertex of the ith copy of H. In this study, we will show that for a graph G with |V(G)| ≥ 2, and a complete graph Kn of order n, n ≤ σ(G ⊙ Kn ) ≤ max {σ(G), n}. In addition, let Pn and Cn denote a path and a cycle of order n respectively. If m, n ≥ 3, we will prove that σ(Km ⊙ Pn ) = 2 if and only if . If n is even, we show that σ(Km ⊙ Cn ) = 2 if and only if . Furthermore, in the case that n is odd, we show that σ(Km ⊙ Cn ) = 3 if and only if where H(r, s) denotes the number of lattice points in the convex hull of points on the plane determined by the integer parameters r and s

d-Lucky Labeling of Graphs

AbstractLet l: V (G) →N be a labeling of the vertices of a graph G by positive integers. Define , where d(u) denotes the degree of u and N(u) denotes the open neighborhood of u. In this paper we introduce a new labeling called d-lucky labeling and study the same as a vertex coloring problem. We define a labeling l as d-lucky if c(u) ≠ c(v) , for every pair of adjacent vertices u and v in G. The d-lucky number of a graph G, denoted by ηdl(G), is the least positive k such that G has a d-lucky labeling with {1,2, ..., k} as the set of labels. We obtain ηdl(G) = 2 for hypercube network, butterfly network, benes network, mesh network, hypertree and X-tree

RG Flows and Fixed Points of O(N)rO(N)^r Models

By means of $\epsilon$ and large $N$ expansions, we study generalizations of the $O(N)$ model where the fundamental fields are tensors of rank $r$ rather than vectors, and where the global symmetry (up to additional discrete symmetries and quotients) is $O(N)^r$, focusing on the cases $r\leq 5$. Owing to the distinct ways of performing index contractions, these theories contain multiple quartic operators, which mix under the RG flow. At all large $N$ fixed points, melonic operators are absent and the leading Feynman diagrams are bubble diagrams, so that all perturbative fixed points can be readily matched to full large $N$ solutions obtained from Hubbard-Stratonovich transformations. The family of fixed points we uncover extend to arbitrary higher values of $r$, and as their number grows superexponentially with $r$, these theories offer a vast generalization of the critical $O(N)$ model. We also study sextic $O(N)^r$ theories, whose large $N$ limits are obscured by the fact that the dominant Feynman diagrams are not restricted to melonic or bubble diagrams. For these theories the large $N$ dynamics differ qualitatively across different values of $r$, and we demonstrate that the RG flows possess a numerous and diverse set of perturbative fixed points beginning at rank four.Comment: 60 pages + appendices and reference

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page