Neighbor product distinguishing total colorings of corona of subcubic graphs

A proper $[k]$-total coloring $c$ of a graph $G$ is a mapping $c$ from $V(G)\bigcup E(G)$ to $[k]=\{1,2,\cdots,k\}$ such that $c(x)\neq c(y)$ for which $x$, $y\in V(G)\bigcup E(G)$ and $x$ is adjacent to or incident with $y$. Let $\prod(v)$ denote the product of $c(v)$ and the colors on all the edges incident with $v$. For each edge $uv\in E(G)$, if $\prod(u)\neq \prod(v)$, then the coloring $c$ is called a neighbor product distinguishing total coloring of $G$. we use $\chi"_{\prod}(G)$ to denote the minimal value of $k$ in such a coloring of $G$. In 2015, Li et al. conjectured that $\Delta(G)+3$ colors enable a graph to have a neighbor product distinguishing total coloring. In this paper, we consider the neighbor product distinguishing total coloring of corona product $G\circ H$, and obtain that $\chi"_{\prod}(G\circ H)\leq \Delta(G\circ H)+3$

Upper Bounds on Sets of Orthogonal Colorings of Graphs

We generalize the notion of orthogonal latin squares to colorings of simple graphs. Two $n$-colorings of a graph are said to be \emph{orthogonal} if whenever two vertices share a color in one coloring they have distinct colors in the other coloring. We show that the usual bounds on the maximum size of a certain set of orthogonal latin structures such as latin squares, row latin squares, equi-$n$ squares, single diagonal latin squares, double diagonal latin squares, or sudoku squares are a special cases of bounds on orthogonal colorings of graphs.Comment: 17 page

Vertex Colorings without Rainbow or Monochromatic Subgraphs

This paper investigates vertex colorings of graphs such that some rainbow subgraph~$R$ and some monochromatic subgraph $M$ are forbidden. Previous work focussed on the case that $R=M$. Here we consider the more general case, especially the case that $M=K_2$

Edge Coloring and Stopping Sets Analysis in Product Codes with MDS components

We consider non-binary product codes with MDS components and their iterative row-column algebraic decoding on the erasure channel. Both independent and block erasures are considered in this paper. A compact graph representation is introduced on which we define double-diversity edge colorings via the rootcheck concept. An upper bound of the number of decoding iterations is given as a function of the graph size and the color palette size $M$. Stopping sets are defined in the context of MDS components and a relationship is established with the graph representation. A full characterization of these stopping sets is given up to a size $(d_1+1)(d_2+1)$, where $d_1$ and $d_2$ are the minimum Hamming distances of the column and row MDS components respectively. Then, we propose a differential evolution edge coloring algorithm that produces colorings with a large population of minimal rootcheck order symbols. The complexity of this algorithm per iteration is $o(M^{\aleph})$, for a given differential evolution parameter $\aleph$, where $M^{\aleph}$ itself is small with respect to the huge cardinality of the coloring ensemble. The performance of MDS-based product codes with and without double-diversity coloring is analyzed in presence of both block and independent erasures. In the latter case, ML and iterative decoding are proven to coincide at small channel erasure probability. Furthermore, numerical results show excellent performance in presence of unequal erasure probability due to double-diversity colorings.Comment: 82 pages, 14 figures, and 4 tables, Submitted to the IEEE Transactions on Information Theory, Dec. 2015, paper IT-15-110

List Coloring a Cartesian Product with a Complete Bipartite Factor

We study the list chromatic number of the Cartesian product of any graph $G$ and a complete bipartite graph with partite sets of size $a$ and $b$, denoted $\chi_\ell(G \square K_{a,b})$. We have two motivations. A classic result on the gap between list chromatic number and the chromatic number tells us $\chi_\ell(K_{a,b}) = 1 + a$ if and only if $b \geq a^a$. Since $\chi_\ell(K_{a,b}) \leq 1 + a$ for any $b \in \mathbb{N}$, this result tells us the values of $b$ for which $\chi_\ell(K_{a,b})$ is as large as possible and far from $\chi(K_{a,b})=2$. In this paper we seek to understand when $\chi_\ell(G \square K_{a,b})$ is far from $\chi(G \square K_{a,b}) = \max \{\chi(G), 2 \}$. It is easy to show $\chi_\ell(G \square K_{a,b}) \leq \chi_\ell (G) + a$. In 2006, Borowiecki, Jendrol, Kr\'al, and Miskuf showed that this bound is attainable if $b$ is sufficiently large; specifically, $\chi_\ell(G \square K_{a,b}) = \chi_\ell (G) + a$ whenever $b \geq (\chi_\ell(G) + a - 1)^{a|V(G)|}$. Given any graph $G$ and $a \in \mathbb{N}$, we wish to determine the smallest $b$ such that $\chi_\ell(G \square K_{a,b}) = \chi_\ell (G) + a$. In this paper we show that the list color function, a list analogue of the chromatic polynomial, provides the right concept and tool for making progress on this problem. Using the list color function, we prove a general improvement on Borowiecki et al.'s 2006 result, and we compute the smallest such $b$ exactly for some large families of chromatic-choosable graphs.Comment: 12 page

Orientations of 1-Factorizations and the List Chromatic Index of Small Graphs

As starting point, we formulate a corollary to the Quantitative Combinatorial Nullstellensatz. This corollary does not require the consideration of any coefficients of polynomials, only evaluations of polynomial functions. In certain situations, our corollary is more directly applicable and more ready-to-go than the Combinatorial Nullstellensatz itself. It is also of interest from a numerical point of view. We use it to explain a well-known connection between the sign of 1-factorizations (edge colorings) and the List Edge Coloring Conjecture. For efficient calculations and a better understanding of the sign, we then introduce and characterize the sign of single 1-factors. We show that the product over all signs of all the 1-factors in a 1-factorization is the sign of that 1-factorization. Using this result in an algorithm, we attempt to prove the List Edge Coloring Conjecture for all graphs with up to 10 vertices. This leaves us with some exceptional cases that need to be attacked with other methods.Comment: 14 page

Qualgebras and knotted 3-valent graphs

This paper is devoted to qualgebras and squandles, which are quandles enriched with a compatible binary/unary operation. Algebraically, they are modeled after groups with conjugation and multiplication/squaring operations. Topologically, qualgebras emerge as an algebraic counterpart of knotted 3-valent graphs, just like quandles can be seen as an "algebraization" of knots; squandles in turn simplify the qualgebra algebraization of graphs. Knotted 3-valent graph invariants are constructed by counting qualgebra/squandle colorings of graph diagrams, and are further enhanced using qualgebra/squandle 2-cocycles. Some algebraic properties and the beginning of a cohomology theory are given for both structures. A classification of size 4 qualgebras/squandles is presented, and their second cohomology groups are completely described

Star Edge-Coloring of Square Grids

A star edge-coloring of a graph $G$ is a proper edge-coloring without bichromatic paths or cycles of length four. The smallest integer $k$ such that $G$ admits a star edge-coloring with $k$ colors is the star chromatic index of $G$. In the seminal paper on the topic, Dvo\v{r}\'{a}k, Mohar, and \v{S}\'{a}mal asked if the star chromatic index of complete graphs is linear in the number of vertices and gave an almost linear upper bound. Their question remains open, and consequently, to better understand the behavior of the star chromatic index, this parameter has been studied for a number of other classes of graphs. In this paper, we consider star edge-colorings of square grids; namely, the Cartesian products of paths and cycles and the Cartesian products of two cycles. We improve previously established bounds and, as a main contribution, we prove that the star chromatic index of graphs in both classes is either $6$ or $7$ except for prisms. Additionally, we give a number of exact values for many considered graphs

Counting without sampling. New algorithms for enumeration problems using statistical physics

We propose a new type of approximate counting algorithms for the problems of enumerating the number of independent sets and proper colorings in low degree graphs with large girth. Our algorithms are not based on a commonly used Markov chain technique, but rather are inspired by developments in statistical physics in connection with correlation decay properties of Gibbs measures and its implications to uniqueness of Gibbs measures on infinite trees, reconstruction problems and local weak convergence methods. On a negative side, our algorithms provide $\epsilon$-approximations only to the logarithms of the size of a feasible set (also known as free energy in statistical physics). But on the positive side, our approach provides deterministic as opposed to probabilistic guarantee on approximations. Moreover, for some regular graphs we obtain explicit values for the counting problem. For example, we show that every 4-regular $n$-node graph with large girth has approximately $(1.494...)^n$ independent sets, and in every $r$-regular graph with $n$ nodes and large girth the number of $q\geq r+1$-proper colorings is approximately $[q(1-{1\over q})^{r\over 2}]^n$, for large $n$. In statistical physics terminology, we compute explicitly the limit of the log-partition function. We extend our results to random regular graphs. Our explicit results would be hard to derive via the Markov chain method.Comment: 23 pages 1 figur

Exact values for the Grundy number of some graphs

The Grundy number of a graph G is the maximum number k of colors used to color the vertices of G such that the coloring is proper and every vertex x colored with color i, is adjacent to (i - 1) vertices colored with each color j, In this paper we give bounds for the Grundy number of some graphs and Cartesian products of graphs. In particular, we determine an exact value of this parameter for n-dimensional meshes and some n-dimensional toroidal meshes. Finally, we present an algorithm to generate all graphs for a given Grundy numbe