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The coming quantum computation is forcing us to reexamine the cryptosystems people use. We are applying graph colorings of topological coding to modern information security and future cryptography against supercomputer and quantum computer…
Making topological authentication from theory to practical application is an important and challenging task. More and more researchers pay attention on coming quantum computation, privacy data protection, lattices and cryptography. Research…
Lattice-based Cryptography is considered to have the characteristics of classical computers and quantum attack resistance. We will design various graphic lattices and matrix lattices based on knowledge of graph theory and topological…
Lattice-based cryptography is not only for thwarting future quantum computers, and is also the basis of Fully Homomorphic Encryption. Motivated from the advantage of graph homomorphisms we combine graph homomorphisms with graph total…
We, for anti-quantum computing, will discuss various number-based strings, such as number-based super-strings, parameterized strings, set-based strings, graph-based strings, integer-partitioned and integer-decomposed strings, Hanzi-based…
In order to make more complex number-based strings from topological coding for defending against the intelligent attacks equipped with quantum computing and providing effective protection technology for the age of quantum computing, we will…
Topological Coding consists of two different kinds of mathematics: topological structure and mathematical relation. The colorings and labelings of graph theory are main techniques in topological coding applied in asymmetric encryption…
We study a graph coloring problem motivated by a fun Sudoku-style puzzle. Given a bipartition of the edges of a graph into {\em near} and {\em far} sets and an integer threshold $t$, a {\em threshold-coloring} of the graph is an assignment…
An odd $k$-edge-coloring of a graph $G$ is a (not necessarily proper) edge-coloring with at most $k$ colors such that each non-empty color class induces a graph in which every vertex is of odd degree; similarly, if more than one color per…
Differential Privacy is the gold standard in privacy-preserving data analysis. This paper addresses the challenge of producing a differentially edge-private vertex coloring. In this paper, we present two novel algorithms to approach this…
The Four color problem is closely related to other branches of mathematics and practical applications. More than 20 of its reformulations are known, which connect this problem with problems of algebra, statistical mechanics and planning.…
We study a combinatorial coloring game between two players, Spoiler and Algorithm, who alternate turns. First, Spoiler places a new token at a vertex in $G$, and Algorithm responds by assigning a color to the new token. Algorithm must…
This paper explores the application of a new algebraic method of color exchanges to the edge coloring of simple graphs. Vizing's theorem states that the edge coloring of a simple graph $G$ requires either $\Delta$ or $\Delta+1$ colors,…
Odd coloring is a variant of proper coloring and has received wide attention. We study the list-coloring version of this notion in this paper. We prove that if $G$ is a graph embeddable in the torus or the Klein bottle with no cycle of…
In the W-streaming model, an algorithm is given $O(n \mathrm{polylog} n)$ space and must process a large graph of up to $O(n^2)$ edges. In this short note we give two algorithms for edge colouring under the W-streaming model. For edge…
In the Coloured Clustering problem, we wish to colour vertices of an edge coloured graph to produce as many stable edges as possible, i.e., edges with the same colour as their ends. In this paper, we reveal that the problem is in fact a…
We obtain several new upper bounds of the odd graceful chromatic number of a graph $G$, which must be bipartite. Some of our bounds depend only on the number of the vertices of $G$ or the chromatic number of some graphs related to the…
Drawings of non-planar graphs always result in edge crossings. When there are many edges crossing at small angles, it is often difficult to follow these edges, because of the multiple visual paths resulted from the crossings that slow down…
Coloring the arcs of biregular graphs was introduced with possible applications to industrial chemistry, molecular biology, cellular neuroscience, etc. Here, we deal with arc coloring in some non-bipartite graphs. In fact, for…
For edge coloring, the online and the W-streaming models seem somewhat orthogonal: the former needs edges to be assigned colors immediately after insertion, typically without any space restrictions, while the latter limits memory to…