Related papers: Some Properties of Fibonacci-Sum Set-Graphs
We study Fibonacci compositions, which are compositions of natural numbers that only use Fibonacci numbers, in two different contexts. We first prove inequalities comparing the number of Fibonacci compositions to regular compositions where…
We review the combinatorial, topological, algebraic and metric properties supported by $(D+1)$-colored graphs, with a focus on those that are pertinent to the study of tensor model theories. We show how to extract a limiting continuum…
In this paper, based on the contributions of Tucker (1983) and Seb{\H{o}} (1992), we generalize the concept of a sequential coloring of a graph to a framework in which the algorithm may use a coloring rule-base obtained from suitable…
The \emph{Square Colouring} of a graph $G$ refers to colouring of vertices of a graph such that any two distinct vertices which are at distance at most two receive different colours. In this paper, we initiate the study of a related…
This paper surveys recent development of concepts related to coloring of signed graphs. Various approaches are presented and discussed.
This paper serves as the first extension of the topic of dominator colorings of graphs to the setting of digraphs. We establish the dominator chromatic number over all possible orientations of paths and cycles. In this endeavor we discover…
A $t$-tone coloring of a graph $G$ assigns to each vertex a set of $t$ colors such that any pair of vertices $u, v$ with distance $d$ can share at most $d-1$ colors. In this note, we prove several new results on $t$-tone coloring. For…
We study the properties of certain graphs involving the sums of primes. Their structure largely turns out to relate to the distribution of prime gaps and can be roughly seen in Cram\'er's model as well. We also discuss generalizations to…
In part 1 of this paper some linear weighted generalized Fibonacci number summation identities were derived using the fact that the Fibonacci number is the residue of a rational function. In this part, using the same method, some quadratic…
In this paper, we consider the problem of a star coloring. In general case the problems in NP-complete. We establish the star chromatic number for splitting graph of complete and complete bipartite graphs, as well of paths and cycles. Our…
This work focuses on the combinatorial properties of glued semigroups and provides its combinatorial characterization. Some classical results for affine glued semigroups are generalized and some methods to obtain glued semigroups are…
In most studies related to colouring of graphs and perhaps in the study of other invariants and variants of graphs, the restrictions of non-triviality and connectedness are placed upon graphs. For the introduction to comp\^{o}nent\u{a}…
The paper investigates relationship between algebraic expressions and graphs. We consider a digraph called a Fibonacci graph which gives a generic example of non-series-parallel graphs. Our intention in this paper is to simplify the…
The symmetric tensor power of graphs is introduced and its fundamental properties are explored. A wide range of intriguing phenomena occur when one considers symmetric tensor powers of familiar graphs. A host of open questions are…
We offer several new summation identities involving harmonic numbers, odd harmonic numbers, and Fibonacci numbers. Our results are derived using three different approaches: partial summation, polynomial identities and binomial…
This paper investigates the coloring problem on Fibonacci-Cayley tree, which is a Cayley graph whose vertex set is the Fibonacci sequence. More precisely, we elucidate the complexity of shifts of finite type defined on Fibonacci-Cayley tree…
Beck's conjecture on coloring of graphs associated to various algebraic objects has generated considerable interest in the community of discrete mathematics and combinatorics since its inception in the year 1988. The version of this…
A colored graph is a directed graph in which nodes or edges have been assigned colors that are not necessarily unique. Observability problems in such graphs consider whether an agent observing the colors of edges or nodes traversed on a…
In this paper, we investigate some properties of generalized Fibonacci quaternions and Fibonacci-Narayana quaternions.
The attempts to prove the Four Color Problem last for long years. A little hope arises that the properties of the minimal partial triangulations will be very useful for the solution of the Four Color Problem. That is why the material of…