Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userKings in Multipartite Hypertournaments
123
0
0.0
(
0
)
Added by
Gregory Gutin
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
In his paper Kings in Bipartite Hypertournaments (Graphs $&$ Combinatorics 35, 2019), Petrovic stated two conjectures on 4-kings in multipartite hypertournaments. We prove one of these conjectures and give counterexamples for the other.
rate research
Read More
An edge-coloring of a graph $G$ with colors $1,ldots,t$ is an emph{interval $t$-coloring} if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an integer interval. It is well-known that there are graphs that do not have interval colorings. The emph{deficiency} of a graph $G$, denoted by $mathrm{def}(G)$, is the minimum number of pendant edges whose attachment to $G$ leads to a graph admitting an interval coloring. In this paper we investigate the problem of determining or bounding of the deficiency of complete multipartite graphs. In particular, we obtain a tight upper bound for the deficiency of complete multipartite graphs. We also determine or bound the deficiency for some classes of complete multipartite graphs.
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 $n+1leq tleq frac{(n+1)(n+2)}{2}$.
Graham and Pollak showed that the vertices of any graph $G$ can be addressed with $N$-tuples of three symbols, such that the distance between any two vertices may be easily determined from their addresses. An addressing is optimal if its length $N$ is minimum possible. In this paper, we determine an addressing of length $k(n-k)$ for the Johnson graphs $J(n,k)$ and we show that our addressing is optimal when $k=1$ or when $k=2, n=4,5,6$, but not when $n=6$ and $k=3$. We study the addressing problem as well as a variation of it in which the alphabet used has more than three symbols, for other graphs such as complete multipartite graphs and odd cycles. We also present computations describing the distribution of the minimum length of addressings for connected graphs with up to $10$ vertices. Motivated by these computations we settle a problem of Graham, showing that most graphs on $n$ vertices have an addressing of length at most $n-(2-o(1))log_2 n$.
In this paper, we extend the concept of kings and serfs in tournaments to that of weak kings and weak serfs in oriented graphs. We obtain various results on the existence of weak kings(weak serfs) in oriented graphs, and show the existence of n-oriented graphs containing exactly k weak kings(weak serfs). Also, we give the existence of n-oriented graphs containing exactly k weak kings and exactly s weak serfs such that b weak kings from k are also weak serfs.
In 1975 Bollobas, ErdH os, and Szemeredi asked the following question: given positive integers $n, t, r$ with $2le tle r-1$, what is the largest minimum degree $delta(G)$ among all $r$-partite graphs $G$ with parts of size $n$ and which do not contain a copy of $K_{t+1}$? The $r=t+1$ case has attracted a lot of attention and was fully resolved by Haxell and Szab{o}, and Szab{o} and Tardos in 2006. In this paper we investigate the $r>t+1$ case of the problem, which has remained dormant for over forty years. We resolve the problem exactly in the case when $r equiv -1 pmod{t}$, and up to an additive constant for many other cases, including when $r geq (3t-1)(t-1)$. Our approach utilizes a connection to the related problem of determining the maximum of the minimum degrees among the family of balanced $r$-partite $rn$-vertex graphs of chromatic number at most $t$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
