اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدSome Results on Cyclic Interval Edge Colorings of Graphs
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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 as consecutive to color $t$. We prove that a bipartite graph $G$ with even maximum degree $Delta(G)geq 4$ admits a cyclic interval $Delta(G)$-coloring if for every vertex $v$ the degree $d_G(v)$ satisfies either $d_G(v)geq Delta(G)-2$ or $d_G(v)leq 2$. We also prove that every Eulerian bipartite graph $G$ with maximum degree at most $8$ has a cyclic interval coloring. Some results are obtained for $(a,b)$-biregular graphs, that is, bipartite graphs with the vertices in one part all having degree $a$ and the vertices in the other part all having degree $b$; it has been conjectured that all these have cyclic interval colorings. We show that all $(4,7)$-biregular graphs as well as all $(2r-2,2r)$-biregular ($rgeq 2$) graphs have cyclic interval colorings. Finally, we prove that all complete multipartite graphs admit cyclic interval colorings; this settles in the affirmative, a conjecture of Petrosyan and Mkhitaryan.
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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_{
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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
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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
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