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Register a new userGraph Homomorphisms Based On Particular Total Colorings of Graphs and Graphic Lattices
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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 colorings together for designing new types of graph homomorphisms: totally-colored graph homomorphisms, graphic-lattice homomorphisms from sets to sets, every-zero graphic group homomorphisms from sets to sets. Our graph-homomorphism lattices are made up by graph homomorphisms. These new homomorphisms induce some problems of graph theory, for example, Number String Decomposition and Graph Homomorphism Problem.
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A total coloring of a graph $G$ is a coloring of its vertices and edges such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color. An interval total $t$-coloring of a graph $G$ is a total coloring of $G$ with colors $1,ldots,t$ such that all colors are used, and the edges incident to each vertex $v$ together with $v$ are colored by $d_{G}(v)+1$ consecutive colors, where $d_{G}(v)$ is the degree of a vertex $v$ in $G$. In this paper we prove that all complete multipartite graphs with the same number of vertices in each part are interval total colorable. Moreover, we also give some bounds for the minimum and the maximum span in interval total colorings of these graphs. Next, we investigate interval total colorings of hypercubes $Q_{n}$. In particular, we prove that $Q_{n}$ ($ngeq 3$) has an interval total $t$-coloring if and only if $n+1leq tleq frac{(n+1)(n+2)}{2}$.
$H_q(n,d)$ is defined as the graph with vertex set ${mathbb Z}_q^n$ and where two vertices are adjacent if their Hamming distance is at least $d$. The chromatic number of these graphs is presented for various sets of parameters $(q,n,d)$. For the $4$-colorings of the graphs $H_2(n,n-1)$ a notion of robustness is introduced. It is based on the tolerance of swapping colors along an edge without destroying properness of the coloring. An explicit description of the maximally robust $4$-colorings of $H_2(n,n-1)$ is presented.
Lattice theory has been believed to resist classical computers and quantum computers. Since there are connections between traditional lattices and graphic lattices, it is meaningful to research graphic lattices. We define the so-called ice-flower systems by our uncolored or colored leaf-splitting and leaf-coinciding operations. These ice-flower systems enable us to construct several star-graphic lattices. We use our star-graphic lattices to express some well-known results of graph theory and compute the number of elements of a particular star-graphic lattice. For more researching ice-flower systems and star-graphic lattices we propose Decomposition Number String Problem, finding strongly colored uniform ice-flower systems and connecting our star-graphic lattices with traditional lattices.
In this paper, we study the achromatic and the pseudoachromatic numbers of planar and outerplanar graphs as well as planar graphs of girth 4 and graphs embedded on a surface. We give asymptotically tight results and lower bounds for maximal embedded graphs.
A proper edge coloring of a graph $G$ with colors $1,2,dots,t$ is called a emph{cyclic interval $t$-coloring} if for each vertex $v$ of $G$ the edges incident to $v$ are colored by consecutive colors, under the condition that color $1$ is considered as consecutive to color $t$. We prove that a bipartite graph $G$ with even maximum degree $Delta(G)geq 4$ admits a cyclic interval $Delta(G)$-coloring if for every vertex $v$ the degree $d_G(v)$ satisfies either $d_G(v)geq Delta(G)-2$ or $d_G(v)leq 2$. We also prove that every Eulerian bipartite graph $G$ with maximum degree at most $8$ has a cyclic interval coloring. Some results are obtained for $(a,b)$-biregular graphs, that is, bipartite graphs with the vertices in one part all having degree $a$ and the vertices in the other part all having degree $b$; it has been conjectured that all these have cyclic interval colorings. We show that all $(4,7)$-biregular graphs as well as all $(2r-2,2r)$-biregular ($rgeq 2$) graphs have cyclic interval colorings. Finally, we prove that all complete multipartite graphs admit cyclic interval colorings; this settles in the affirmative, a conjecture of Petrosyan and Mkhitaryan.
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