arXiv subjects

Yehuda Roditty

Publications and source records attributed to Yehuda Roditty.

4 recordsLinked to original sources

arXiv2023

On the class reconstruction number of trees

Harary and Lauri conjectured that the class reconstruction number of trees is 2, that is, each tree has two unlabelled vertex-deleted subtrees that are not both in the deck of any other tree. We show that each tree $T$ can be reconstructed up to isomorphism given two of its unlabelled subgraphs $T-u$ and $T-v$ under the assumption that $u$ and $v$ are chosen in a particular way. Our result does not completely resolve the conjecture of Harary and Lauri since the special property defining $u$ and $v$ cannot be recognised from the given subtrees $T-u$ and $T-v$.

math.CO↗

arXiv2015

Line-Broadcasting in Complete k-Trees

A line-broadcasting model in a connected graph $G=(V,E)$, $|V|=n$, is a model in which one vertex, called the {\it originator} of the broadcast holds a message that has to be transmitted to all vertices of the graph through placement of a series of calls over the graph. In this model, an informed vertex can transmit a message through a path of any length in a single time unit, as long as two transmissions do not use the same edge at the same time. Farley \cite{f} has shown that the process is completed within at most $\lceil \log_{2}n \rceil$ time units from any originator in a tree (and thus in any connected undirected graph). and that the cost of broadcasting one message from any vertex is at most $(n-1) \lceil \log_{2}n \rceil$. In this paper, we present lower and upper bounds for the cost to broadcast one message in a complete $k-$tree, from any vertex using the line-broadcasting model. We prove that if $B(u)$ is the minimum cost to broadcast in a graph $G=(V,E)$ from a vertex $u \in V$ using the line-broadcasting model, then $(1+o(1))n \le B(u) \le (2+o(1))n$, where $u$ is any vertex in a complete $k$-tree. Furthermore, for certain conditions, $B(u) \le (2-o(1))n$.

cs.DM↗

arXiv2013

Anti-Ramsey numbers of small graphs

The anti-Ramsey number $AR(n,G$), for a graph $G$ and an integer $n\geq|V(G)|$, is defined to be the minimal integer $r$ such that in any edge-colouring of $K_n$ by at least $r$ colours there is a multicoloured copy of $G$, namely, a copy of $G$ whose edges have distinct colours. In this paper we determine the anti-Ramsey numbers of all graphs having at most four edges.

math.CO↗

arXiv2013

Anti-Ramsey numbers of graphs with small connected components

The anti-Ramsey number, $AR(n,G)$, for a graph $G$ and an integer $n\geq|V(G)|$, is defined to be the minimal integer $r$ such that in any edge-colouring of $K_n$ by at least $r$ colours there is a multicoloured copy of $G$, namely, a copy of $G$ that each of its edges has a distinct colour. In this paper we determine, for large enough $n$, $AR(n,L\cup tP_2)$ and $AR(n,L\cup kP_3)$ for any large enough $t$ and $k$, and a graph $L$ satisfying some conditions. Consequently, we determine $AR(n,G)$, for large enough $n$, where $G$ is $P_3\cup tP_2$ for any $t\geq 3$, $P_4\cup tP_2$ and $C_3\cup tP_2$ for any $t\geq 2$, $kP_3$ for any $k\geq 3$, $tP_2\cup kP_3$ for any $t\geq 1$, $k\geq 2$, and $P_{t+1}\cup kP_3$ for any $t\geq 3$, $k\geq 1$. Furthermore, we obtain upper and lower bounds for $AR(n,G)$, for large enough $n$, where $G$ is $P_{k+1}\cup tP_2$ and $C_k\cup tP_2$ for any $k\geq 4$, $t\geq 1$.

math.CO↗