An ℓ-facial edge-coloring of a plane graph is coloring its edges such that any two at distance most ℓ on boundary walk face receive distinct colors. It the variant vertex coloring, which arose as generalization well-known cyclic coloring. conjectured 3ℓ+1 colors suffice for an graph. The conjecture has only been confirmed ℓ≤2, and in this paper, we prove validity ℓ=3.

Clustering algorithms consist in automatically discovering structure of large data set or providing coherent groups of objects independent of any user-defined classes. They are used in many domains: astronomy, information retrieval, image segmentation, biological applications and so on. Several clustering techniques have been proposed [1], there are either hierarchical techniques or partitionin...

In this paper, author uses set theory to construct a logic model of abstract figure from binary relation. Based on the uniform quantified structure, author gives two logic system for graph traversal and graph coloring respectively, moreover shows a new method of cutting graph. Around this model, there are six algorithms in this paper including exact graph traversal, Algebra calculation of natur...

In this paper an improved bound on the chromatic number of Pancake graph $P_n, n\geqslant 2$, is presented. The obtained using a subadditivity property graph. We also investigate equitable coloring $P_n$. An $(n-1)$-coloring based efficient dominating sets given and optimal $4$-colorings are considered for small $n$. It conjectured that $P_n$ coincides with its any $n\geqslant 2$.

in the present study, a simple and sensitive extractive spectrophotometric method is described for determination of buspirone. the method is based on the reaction of buspirone and bromocresol green. the ion-pair complex was quantitatively extracted into chloroform at ph 2.3 followed by spectrophotometric determination at 415 nm. the complex was stable up to 2 days and obeyed beer's law over the...

a simple and sensitive extractive spectrophotometric method is described for determination of tropicamide. the method is based on the reaction of tropicamide and bromocresol green. the ion-paired colored complex was extracted with chloroform at ph 3. the extracted complex showed maximum absorbance at 423 nm. the complex was stable up to 2 days and obeyed beer's law over the concentration r...

the colorful paths and rainbow paths have been considered by severalauthors.a colorful directed path in a digraph $g$ is a directed path with $chi(g)$ vertices whose colors are different. a $v$-colorful directed path is such a directed path, starting from $v$. we prove that for a given $3$-regular triangle-free digraph $g$ determining whether there is a proper $chi(g)$-coloring of $g$such that ...