اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدLower bounds for the greatest possible number of colors in interval edge colorings of bipartite cylinders and bipartite tori
98
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
An interval edge t-coloring of a graph G is a proper edge coloring of G with colors 1,2...,t such that at least one edge of G is colored by color i,i=1,2...,t, and the edges incident with each vertex v are colored by d_{G}(v) consecutive colors, where d_{G}(v) is the degree of the vertex v in G. In this paper interval edge colorings of bipartite cylinders and bipartite tori are investigated.
قيم البحث
اقرأ أيضاً
An emph{interval $t$-coloring} of a multigraph $G$ is a proper edge coloring with colors $1,dots,t$ such that the colors on the edges incident to every vertex of $G$ are colored by consecutive colors. A emph{cyclic interval $t$-coloring} of a multigr
aph $G$ is a proper edge coloring with colors $1,dots,t$ such that the colors on the edges incident to every vertex of $G$ are colored by consecutive colors, under the condition that color $1$ is considered as consecutive to color $t$. Denote by $w(G)$ ($w_{c}(G)$) and $W(G)$ ($W_{c}(G)$) the minimum and maximum number of colors in a (cyclic) interval coloring of a multigraph $G$, respectively. We present some new sharp bounds on $w(G)$ and $W(G)$ for multigraphs $G$ satisfying various conditions. In particular, we show that if $G$ is a $2$-connected multigraph with an interval coloring, then $W(G)leq 1+leftlfloor frac{|V(G)|}{2}rightrfloor(Delta(G)-1)$. We also give several results towards the general conjecture that $W_{c}(G)leq |V(G)|$ for any triangle-free graph $G$ with a cyclic interval coloring; we establish that approximat
An edge-coloring of a graph $G$ with colors $1,2,ldots,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable if i
t has an interval $t$-coloring for some positive integer $t$. For an interval colorable graph $G$, $W(G)$ denotes the greatest value of $t$ for which $G$ has an interval $t$-coloring. It is known that the complete graph is interval colorable if and only if the number of its vertices is even. However, the exact value of $W(K_{2n})$ is known only for $n leq 4$. The second author showed that if $n = p2^q$, where $p$ is odd and $q$ is nonnegative, then $W(K_{2n}) geq 4n-2-p-q$. Later, he conjectured that if $n in mathbb{N}$, then $W(K_{2n}) = 4n - 2 - leftlfloorlog_2{n}rightrfloor - left | n_2 right |$, where $left | n_2 right |$ is the number of $1$s in the binary representation of $n$. In this paper we introduce a new technique to construct interval colorings of complete graphs based on their 1-factorizations, which is used to disprove the conjecture, improve lower and upper bounds on $W(K_{2n})$ and determine its exact values for $n leq 12$.
A $k$-improper edge coloring of a graph $G$ is a mapping $alpha:E(G)longrightarrow mathbb{N}$ such that at most $k$ edges of $G$ with a common endpoint have the same color. An improper edge coloring of a graph $G$ is called an improper interval edge
coloring if the colors of the edges incident to each vertex of $G$ form an integral interval. In this paper we introduce and investigate a new notion, the interval coloring impropriety (or just impropriety) of a graph $G$ defined as the smallest $k$ such that $G$ has a $k$-improper interval edge coloring; we denote the smallest such $k$ by $mu_{mathrm{int}}(G)$. We prove upper bounds on $mu_{mathrm{int}}(G)$ for general graphs $G$ and for particular families such as bipartite, complete multipartite and outerplanar graphs; we also determine $mu_{mathrm{int}}(G)$ exactly for $G$ belonging to some particular classes of graphs. Furthermore, we provide several families of graphs with large impropriety; in particular, we prove that for each positive integer $k$, there exists a graph $G$ with $mu_{mathrm{int}}(G) =k$. Finally, for graphs with at least two vertices we prove a new upper bound on the number of colors used in an improper interval edge coloring.
A proper edge-coloring of a graph $G$ with colors $1,ldots,t$ is called an emph{interval cyclic $t$-coloring} if all colors are used, and the edges incident to each vertex $vin V(G)$ are colored by $d_{G}(v)$ consecutive colors modulo $t$, where $d_{
G}(v)$ is the degree of a vertex $v$ in $G$. A graph $G$ is emph{interval cyclically colorable} if it has an interval cyclic $t$-coloring for some positive integer $t$. The set of all interval cyclically colorable graphs is denoted by $mathfrak{N}_{c}$. For a graph $Gin mathfrak{N}_{c}$, the least and the greatest values of $t$ for which it has an interval cyclic $t$-coloring are denoted by $w_{c}(G)$ and $W_{c}(G)$, respectively. In this paper we investigate some properties of interval cyclic colorings. In particular, we prove that if $G$ is a triangle-free graph with at least two vertices and $Gin mathfrak{N}_{c}$, then $W_{c}(G)leq vert V(G)vert +Delta(G)-2$. We also obtain bounds on $w_{c}(G)$ and $W_{c}(G)$ for various classes of graphs. Finally, we give some methods for constructing of interval cyclically non-colorable graphs.
An edge-coloring of a graph $G$ with consecutive integers $c_{1},ldots,c_{t}$ is called an emph{interval $t$-coloring} if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. A grap
h $G$ is interval colorable if it has an interval $t$-coloring for some positive integer $t$. The set of all interval colorable graphs is denoted by $mathfrak{N}$. In 2004, Giaro and Kubale showed that if $G,Hin mathfrak{N}$, then the Cartesian product of these graphs belongs to $mathfrak{N}$. In the same year they formulated a similar problem for the composition of graphs as an open problem. Later, in 2009, the first author showed that if $G,Hin mathfrak{N}$ and $H$ is a regular graph, then $G[H]in mathfrak{N}$. In this paper, we prove that if $Gin mathfrak{N}$ and $H$ has an interval coloring of a special type, then $G[H]in mathfrak{N}$. Moreover, we show that all regular graphs, complete bipartite graphs and trees have such a special interval coloring. In particular, this implies that if $Gin mathfrak{N}$ and $T$ is a tree, then $G[T]in mathfrak{N}$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
