ترغب بنشر مسار تعليمي؟ اضغط هنا

On Equivalent Color Transform and Four Coloring Theorem

75   0   0.0 ( 0 )
 نشر من قبل David Tian
 تاريخ النشر 2016
  مجال البحث الهندسة المعلوماتية
والبحث باللغة English
 تأليف Wenhong Tian




اسأل ChatGPT حول البحث

In this paper, we apply an equivalent color transform (ECT) for a minimal $k$-coloring of any graph $G$. It contracts each color class of the graph to a single vertex and produces a complete graph $K_k$ for $G$ by removing redundant edges between any two vertices. Based on ECT, a simple proof for four color theorem for planar graph is then proposed.

قيم البحث

اقرأ أيضاً

In [J. Combin. Theory Ser. B 70 (1997), 2-44] we gave a simplified proof of the Four-Color Theorem. The proof is computer-assisted in the sense that for two lemmas in the article we did not give proofs, and instead asserted that we have verified thos e statements using a computer. Here we give additional details for one of those lemmas, and we include the original computer programs and data as ancillary files accompanying this submission.
74 - X.-J. Wang , T.-Q. Wang 2021
For the four-color theorem that has been developed over one and half centuries, all people believe it right but without complete proof convincing all1-3. Former proofs are to find the basic four-colorable patterns on a planar graph to reduce a map co loring4-6, but the unavoidable set is almost limitless and required recoloring hardly implements by hand7-14. Another idea belongs to formal proof limited to logical operation15. However, recoloring or formal proof way may block people from discovering the inherent essence of a coloring graph. Defining creation and annihilation operations, we show that four colors are sufficient to color a map and how to color it. We find what trapped vertices and boundary-vertices are, and how they decide how many colors to be required in coloring arbitrary maps. We reveal that there is the fourth color for new adding vertex differing from any three coloring vertices in creation operation. To implement a coloring map, we also demonstrate how to color an arbitrary map by iteratively using creation and annihilation operations. We hope our hand proof is beneficial to understand the mechanisms of the four-color theorem.
Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the parameterized complexity of the following problems parameterized by $k$: (1) Given a graph $G$, a clique modulator $D$ (a clique modulator is a set of vertices, whose removal resu lts in a clique) of size $k$ for $G$, and a list $L(v)$ of colors for every $vin V(G)$, decide whether $G$ has a proper list coloring; (2) Given a graph $G$, a clique modulator $D$ of size $k$ for $G$, and a pre-coloring $lambda_P: X rightarrow Q$ for $X subseteq V(G),$ decide whether $lambda_P$ can be extended to a proper coloring of $G$ using only colors from $Q.$ For Problem 1 we design an $O^*(2^k)$-time randomized algorithm and for Problem 2 we obtain a kernel with at most $3k$ vertices. Banik et al. (IWOCA 2019) proved the the following problem is fixed-parameter tractable and asked whether it admits a polynomial kernel: Given a graph $G$, an integer $k$, and a list $L(v)$ of exactly $n-k$ colors for every $v in V(G),$ decide whether there is a proper list coloring for $G.$ We obtain a kernel with $O(k^2)$ vertices and colors and a compression to a variation of the problem with $O(k)$ vertices and $O(k^2)$ colors.
We show that the mathematical proof of the four color theorem yields a perfect interpretation of the Standard Model of particle physics. The steps of the proof enable us to construct the t-Riemann surface and particle frame which forms the gauge. We specify well-defined rules to match the Standard Model in a one-to-one correspondence with the topological and algebraic structure of the particle frame. This correspondence is exact - it only allows the particles and force fields to have the observable properties of the Standard Model, giving us a Grand Unified Theory. In this paper, we concentrate on explicitly specifying the quarks, gauge vector bosons, the Standard Model scalar Higgs $H^{0}$ boson and the weak force field. Using all the specifications of our mathematical model, we show how to calculate the values of the Weinberg and Cabibbo angles on the particle frame. Finally, we present our prediction of the Higgs $H^{0}$ boson mass $M_{H^{0}} = 125.992 simeq 126 GeV$, as a direct consequence of the proof of the four color theorem.
We consider a decentralized graph coloring model where each vertex only knows its own color and whether some neighbor has the same color as it. The networking community has studied this model extensively due to its applications to channel selection, rate adaptation, etc. Here, we analyze variants of a simple algorithm of Bhartia et al. [Proc., ACM MOBIHOC, 2016]. In particular, we introduce a variant which requires only $O(nlogDelta)$ expected recolorings that generalizes the coupon collector problem. Finally, we show that the $O(nDelta)$ bound Bhartia et al. achieve for their algorithm still holds and is tight in adversarial scenarios.
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