ON LOCAL IRREGULARITY OF THE VERTEX COLORING OF THE CORONA PRODUCT OF A TREE GRAPH

Let $G=(V,E)$ be a graph with a vertex set $V$ and an edge set $E$. The graph $G$ is said to be with a local irregular vertex coloring if there is a function $f$ called a local irregularity vertex coloring with the properties: (i) $l:(V(G)) \to \{ 1,2,...,k \}$ as a vertex irregular $k$-labeling and $w:V(G)\to N,$ for every $uv \in E(G),$ ${w(u)\neq w(v)}$ where $w(u)=\sum_{v\in N(u)}l(i)$ and  (ii) $\mathrm{opt}(l)=\min\{ \max \{ l_{i}:  l_{i} \ \text{is a vertex irregular labeling}\}\}$. The chromatic number of the local irregularity vertex coloring of $G$ denoted by $\chi_{lis}(G)$, is the minimum cardinality of the largest label over all such local irregularity vertex colorings. In this paper, we study a local irregular vertex coloring of $P_m\bigodot G$ when $G$ is a family of tree graphs, centipede $C_n$, double star graph $(S_{2,n})$, Weed graph $(S_{3,n})$, and $E$ graph $(E_{3,n})$

Group twin coloring of graphs

For a given graph $G$, the least integer $k\geq 2$ such that for every Abelian group $\mathcal{G}$ of order $k$ there exists a proper edge labeling $f:E(G)\rightarrow \mathcal{G}$ so that $\sum_{x\in N(u)}f(xu)\neq \sum_{x\in N(v)}f(xv)$ for each edge $uv\in E(G)$ is called the \textit{group twin chromatic index} of $G$ and denoted by $\chi'_g(G)$. This graph invariant is related to a few well-known problems in the field of neighbor distinguishing graph colorings. We conjecture that $\chi'_g(G)\leq \Delta(G)+3$ for all graphs without isolated edges, where $\Delta(G)$ is the maximum degree of $G$, and provide an infinite family of connected graph (trees) for which the equality holds. We prove that this conjecture is valid for all trees, and then apply this result as the base case for proving a general upper bound for all graphs $G$ without isolated edges: $\chi'_g(G)\leq 2(\Delta(G)+{\rm col}(G))-5$, where ${\rm col}(G)$ denotes the coloring number of $G$. This improves the best known upper bound known previously only for the case of cyclic groups $\mathbb{Z}_k$

Connectivity Compression for Irregular Quadrilateral Meshes

Applications that require Internet access to remote 3D datasets are often limited by the storage costs of 3D models. Several compression methods are available to address these limits for objects represented by triangle meshes. Many CAD and VRML models, however, are represented as quadrilateral meshes or mixed triangle/quadrilateral meshes, and these models may also require compression. We present an algorithm for encoding the connectivity of such quadrilateral meshes, and we demonstrate that by preserving and exploiting the original quad structure, our approach achieves encodings 30 - 80% smaller than an approach based on randomly splitting quads into triangles. We present both a code with a proven worst-case cost of 3 bits per vertex (or 2.75 bits per vertex for meshes without valence-two vertices) and entropy-coding results for typical meshes ranging from 0.3 to 0.9 bits per vertex, depending on the regularity of the mesh. Our method may be implemented by a rule for a particular splitting of quads into triangles and by using the compression and decompression algorithms introduced in [Rossignac99] and [Rossignac&Szymczak99]. We also present extensions to the algorithm to compress meshes with holes and handles and meshes containing triangles and other polygons as well as quads

Message passing for the coloring problem: Gallager meets Alon and Kahale

Message passing algorithms are popular in many combinatorial optimization problems. For example, experimental results show that {\em survey propagation} (a certain message passing algorithm) is effective in finding proper $k$-colorings of random graphs in the near-threshold regime. In 1962 Gallager introduced the concept of Low Density Parity Check (LDPC) codes, and suggested a simple decoding algorithm based on message passing. In 1994 Alon and Kahale exhibited a coloring algorithm and proved its usefulness for finding a $k$-coloring of graphs drawn from a certain planted-solution distribution over $k$-colorable graphs. In this work we show an interpretation of Alon and Kahale's coloring algorithm in light of Gallager's decoding algorithm, thus showing a connection between the two problems - coloring and decoding. This also provides a rigorous evidence for the usefulness of the message passing paradigm for the graph coloring problem. Our techniques can be applied to several other combinatorial optimization problems and networking-related issues.Comment: 11 page

The Complexity of Distributed Edge Coloring with Small Palettes

The complexity of distributed edge coloring depends heavily on the palette size as a function of the maximum degree $\Delta$. In this paper we explore the complexity of edge coloring in the LOCAL model in different palette size regimes. 1. We simplify the \emph{round elimination} technique of Brandt et al. and prove that $(2\Delta-2)$-edge coloring requires $\Omega(\log_\Delta \log n)$ time w.h.p. and $\Omega(\log_\Delta n)$ time deterministically, even on trees. The simplified technique is based on two ideas: the notion of an irregular running time and some general observations that transform weak lower bounds into stronger ones. 2. We give a randomized edge coloring algorithm that can use palette sizes as small as $\Delta + \tilde{O}(\sqrt{\Delta})$, which is a natural barrier for randomized approaches. The running time of the algorithm is at most $O(\log\Delta \cdot T_{LLL})$, where $T_{LLL}$ is the complexity of a permissive version of the constructive Lovasz local lemma. 3. We develop a new distributed Lovasz local lemma algorithm for tree-structured dependency graphs, which leads to a $(1+\epsilon)\Delta$-edge coloring algorithm for trees running in $O(\log\log n)$ time. This algorithm arises from two new results: a deterministic $O(\log n)$-time LLL algorithm for tree-structured instances, and a randomized $O(\log\log n)$-time graph shattering method for breaking the dependency graph into independent $O(\log n)$-size LLL instances. 4. A natural approach to computing $(\Delta+1)$-edge colorings (Vizing's theorem) is to extend partial colorings by iteratively re-coloring parts of the graph. We prove that this approach may be viable, but in the worst case requires recoloring subgraphs of diameter $\Omega(\Delta\log n)$. This stands in contrast to distributed algorithms for Brooks' theorem, which exploit the existence of $O(\log_\Delta n)$-length augmenting paths