اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدImproper interval edge colorings of graphs
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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.
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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 $G[T]in mathfrak{N}$.
A proper edge coloring of a graph $G$ with colors $1,2,dots,t$ is called a emph{cyclic interval $t$-coloring} if for each vertex $v$ of $G$ the edges incident to $v$ are colored by consecutive colors, under the condition that color $1$ is considered
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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
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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
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