ترغب بنشر مسار تعليمي؟ اضغط هنا

The $k$-proper index of graphs

103   0   0.0 ( 0 )
 نشر من قبل Xueliang Li
 تاريخ النشر 2016
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

A tree $T$ in an edge-colored graph is a emph{proper tree} if any two adjacent edges of $T$ are colored with different colors. Let $G$ be a graph of order $n$ and $k$ be a fixed integer with $2leq kleq n$. For a vertex set $Ssubseteq V(G)$, a tree containing the vertices of $S$ in $G$ is called an emph{$S$-tree}. An edge-coloring of $G$ is called a emph{$k$-proper coloring} if for every set $S$ of $k$ vertices in $G$, there exists a proper $S$-tree in $G$. The emph{$k$-proper index} of a nontrivial connected graph $G$, denoted by $px_k(G)$, is the smallest number of colors needed in a $k$-proper coloring of $G$. In this paper, some simple observations about $px_k(G)$ for a nontrivial connected graph $G$ are stated. Meanwhile, the $k$-proper indices of some special graphs are determined, and for every pair of positive integers $a$, $b$ with $2leq aleq b$, a connected graph $G$ with $px_k(G)=a$ and $rx_k(G)=b$ is constructed for each integer $k$ with $3leq kleq n$. Also, the graphs with $k$-proper index $n-1$ and $n-2$ are respectively characterized.



قيم البحث

اقرأ أيضاً

111 - Fabio Botler , Lucas Colucci , 2020
Let $chi_k(G)$ denote the minimum number of colors needed to color the edges of a graph $G$ in a way that the subgraph spanned by the edges of each color has all degrees congruent to $1 pmod k$. Scott [{em Discrete Math. 175}, 1-3 (1997), 289--291] p roved that $chi_k(G)leq5k^2log k$, and thus settled a question of Pyber [{em Sets, graphs and numbers} (1992), pp. 583--610], who had asked whether $chi_k(G)$ can be bounded solely as a function of $k$. We prove that $chi_k(G)=O(k)$, answering affirmatively a question of Scott.
Let $G$ be a nonempty simple graph with a vertex set $V(G)$ and an edge set $E(G)$. For every injective vertex labeling $f:V(G)tomathbb{Z}$, there are two induced edge labelings, namely $f^+:E(G)tomathbb{Z}$ defined by $f^+(uv)=f(u)+f(v)$, and $f^-:E (G)tomathbb{Z}$ defined by $f^-(uv)=|f(u)-f(v)|$. The sum index and the difference index are the minimum cardinalities of the ranges of $f^+$ and $f^-$, respectively. We provide upper and lower bounds on the sum index and difference index, and determine the sum index and difference index of various families of graphs. We also provide an interesting conjecture relating the sum index and the difference index of graphs.
The Wiener index of a graph $G$, denoted $W(G)$, is the sum of the distances between all pairs of vertices in $G$. E. Czabarka, et al. conjectured that for an $n$-vertex, $ngeq 4$, simple quadrangulation graph $G$, begin{equation*}W(G)leq begin{cas es} frac{1}{12}n^3+frac{7}{6}n-2, &text{ $nequiv 0~(mod 2)$,} frac{1}{12}n^3+frac{11}{12}n-1, &text{ $nequiv 1~(mod 2)$}. end{cases} end{equation*} In this paper, we confirm this conjecture.
A connected graph $G$ is said to be $k$-connected if it has more than $k$ vertices and remains connected whenever fewer than $k$ vertices are deleted. In this paper, for a connected graph $G$ with sufficiently large order, we present a tight sufficie nt condition for $G$ with fixed minimum degree to be $k$-connected based on the $Q$-index. Our result can be viewed as a spectral counterpart of the corresponding Dirac type condition.
Let $Sz(G),Sz^*(G)$ and $W(G)$ be the Szeged index, revised Szeged index and Wiener index of a graph $G.$ In this paper, the graphs with the fourth, fifth, sixth and seventh largest Wiener indices among all unicyclic graphs of order $ngeqslant 10$ ar e characterized; as well the graphs with the first, second, third, and fourth largest Wiener indices among all bicyclic graphs are identified. Based on these results, further relation on the quotients between the (revised) Szeged index and the Wiener index are studied. Sharp lower bound on $Sz(G)/W(G)$ is determined for all connected graphs each of which contains at least one non-complete block. As well the connected graph with the second smallest value on $Sz^*(G)/W(G)$ is identified for $G$ containing at least one cycle.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا