Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFixing Numbers of Graphs and Groups
79
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The fixing number of a graph $G$ is the smallest cardinality of a set of vertices $S$ such that only the trivial automorphism of $G$ fixes every vertex in $S$. The fixing set of a group $Gamma$ is the set of all fixing numbers of finite graphs with automorphism group $Gamma$. Several authors have studied the distinguishing number of a graph, the smallest number of labels needed to label $G$ so that the automorphism group of the labeled graph is trivial. The fixing number can be thought of as a variation of the distinguishing number in which every label may be used only once, and not every vertex need be labeled. We characterize the fixing sets of finite abelian groups, and investigate the fixing sets of symmetric groups.
rate research
Read More
An edge-coloring of a connected graph $G$ is called a {em monochromatic connection coloring} (MC-coloring for short) if any two vertices of $G$ are connected by a monochromatic path in $G$. For a connected graph $G$, the {em monochromatic connection number} (MC-number for short) of $G$, denoted by $mc(G)$, is the maximum number of colors that ensure $G$ has a monochromatic connection coloring by using this number of colors. This concept was introduced by Caro and Yuster in 2011. They proved that $mc(G)leq m-n+k$ if $G$ is not a $k$-connected graph. In this paper we depict all graphs with $mc(G)=m-n+k+1$ and $mc(G)= m-n+k$ if $G$ is a $k$-connected but not $(k+1)$-connected graph. We also prove that $mc(G)leq m-n+4$ if $G$ is a planar graph, and classify all planar graphs by their monochromatic connectivity numbers.
We define three new pebbling parameters of a connected graph $G$, the $r$-, $g$-, and $u$-critical pebbling numbers. Together with the pebbling number, the optimal pebbling number, the number of vertices $n$ and the diameter $d$ of the graph, this yields 7 graph parameters. We determine the relationships between these parameters. We investigate properties of the $r$-critical pebbling number, and distinguish between greedy graphs, thrifty graphs, and graphs for which the $r$-critical pebbling number is $2^d$.
The theta graph $Theta_{ell,t}$ consists of two vertices joined by $t$ vertex-disjoint paths of length $ell$ each. For fixed odd $ell$ and large $t$, we show that the largest graph not containing $Theta_{ell,t}$ has at most $c_{ell} t^{1-1/ell}n^{1+1/ell}$ edges and that this is tight apart from the value of $c_{ell}$.
In this paper, we consider a variant of Ramsey numbers which we call complementary Ramsey numbers $bar{R}(m,t,s)$. We first establish their connections to pairs of Ramsey $(s,t)$-graphs. Using the classification of Ramsey $(s,t)$-graphs for small $s,t$, we determine the complementary Ramsey numbers $bar{R}(m,t,s)$ for $(s,t)=(4,4)$ and $(3,6)$.
A piecewise linear curve in the plane made up of $k+1$ line segments, each of which is either horizontal or vertical, with consecutive segments being of different orientation is called a $k$-bend path. Given a graph $G$, a collection of $k$-bend paths in which each path corresponds to a vertex in $G$ and two paths have a common point if and only if the vertices corresponding to them are adjacent in $G$ is called a $B_k$-VPG representation of $G$. Similarly, a collection of $k$-bend paths each of which corresponds to a vertex in $G$ is called an $B_k$-EPG representation of $G$ if any two paths have a line segment of non-zero length in common if and only if their corresponding vertices are adjacent in $G$. The VPG bend-number $b_v(G)$ of a graph $G$ is the minimum $k$ such that $G$ has a $B_k$-VPG representation. Similarly, the EPG bend-number $b_e(G)$ of a graph $G$ is the minimum $k$ such that $G$ has a $B_k$-EPG representation. Halin graphs are the graphs formed by taking a tree with no degree $2$ vertex and then connecting its leaves to form a cycle in such a way that the graph has a planar embedding. We prove that if $G$ is a Halin graph then $b_v(G) leq 1$ and $b_e(G) leq 2$. These bounds are tight. In fact, we prove the stronger result that if $G$ is a planar graph formed by connecting the leaves of any tree to form a simple cycle, then it has a VPG-representation using only one type of 1-bend paths and an EPG-representation using only one type of 2-bend paths.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
