In graph theory, a total coloring is a coloring on the vertices and edges of a graph such that: (1). no adjacent vertices have the same color; (2). no adjacent edges have the same color; and (3). no edge and its endvertices are assigned the same color. In 2005, Zhang et al. added a restriction to the definition of total coloring and proposed a new type of coloring defined as follows. Let G = (V,E) be a simple graph endowed with a total coloring φ, and let u be a vertex of G. The set of colors that occurs in the vertex u is defined as C(u) = {φ(u)} ∪ {φ(uv) | uv ∈ E(G)}. Two vertices u,v ∈ V(G) are distinguishable if their color-sets are distinct, i.e., C(u) ≠ C(v). In graph theory, a total coloring is an adjacent-vertex-distinguishing-total-coloring (AVD-total-coloring) if it has the following additional property: (4). for every two adjacent vertices u,v of a graph G, their colors-sets are distinct from each other, i.e., C(u) ≠ C(v). The adjacent-vertex-distinguishing-total-chromatic number χat(G) of a graph G is the fewest colors needed in an AVD-total-coloring of G. The following lower bound for the AVD-total chromatic number can be obtained from the definition of AVD-total-coloring: If a simple graph G has two adjacent vertices of maximum degree, then χat(G) ≥ Δ(G) + 2. Otherwise, if a simple graph G does not have two adjacent vertices of maximum degree, then χat(G) ≥ Δ(G) + 1. In 2005, Zhang et al. determined the AVD-total-chromatic number for some classes of graphs, and based in their results they conjectured the following. AVD-Total-Coloring Conjecture. (Zhang et al.) χat(G) ≤ Δ(G) + 3. The AVD-Total-Coloring Conjecture is known to hold for some classes of graphs, such as complete graphs, graphs with Δ=3, and all bipartite graphs. In 2012, Huang et al. showed that χat(G) ≤ 2Δ(G)for any simple graph G with maximum degree Δ(G) > 2.In 2014, Papaioannou and Raftopoulou described an algorithmic procedure that gives a 7-AVD-total-colouring for any 4-regular graph. (Wikipedia).

definition of adjacent angles

From playlist Common Core Standards - 8th Grade

What are Adjacent Vertices? | Graph Theory

What are adjacent vertices? Graphs are made up of vertices and edges. Each edge joins a pair of vertices. When two vertices are joined by an edge, we say those vertices are "adjacent". Adjacent vertices are vertices that are joined by an edge, non-adjacent vertices are vertices not joined

From playlist Graph Theory

Watch more videos on http://www.brightstorm.com/math/geometry SUBSCRIBE FOR All OUR VIDEOS! https://www.youtube.com/subscription_center?add_user=brightstorm2 VISIT BRIGHTSTORM.com FOR TONS OF VIDEO TUTORIALS AND OTHER FEATURES! http://www.brightstorm.com/ LET'S CONNECT! Facebook ► http

From playlist Geometry

Watch more videos on http://www.brightstorm.com/math/geometry SUBSCRIBE FOR All OUR VIDEOS! https://www.youtube.com/subscription_center?add_user=brightstorm2 VISIT BRIGHTSTORM.com FOR TONS OF VIDEO TUTORIALS AND OTHER FEATURES! http://www.brightstorm.com/ LET'S CONNECT! Facebook ► https

From playlist Geometry

Chromatic Number of Complete Graphs | Graph Theory

What are the chromatic numbers of complete graphs on n vertices? As we’ll see in today’s graph theory lesson on vertex coloring, we need exactly n colors to properly color the complete graph K_n. Intro to Graph Colorings: https://youtu.be/3VeQhNF5-rE Recall that a proper coloring (or ju

From playlist Graph Theory

Geometry - How to show two triangles are similar using AA with parallel sides

👉 Learn how to solve with similar triangles. Two triangles are said to be similar if the corresponding angles are congruent (equal). Note that two triangles are similar does not imply that the length of the sides are equal but the sides are proportional. Knowledge of the length of the side

From playlist Similar Triangles

Angle Bisectors and Opposite Side Ratios

Watch more videos on http://www.brightstorm.com/math/geometry SUBSCRIBE FOR All OUR VIDEOS! https://www.youtube.com/subscription_center?add_user=brightstorm2 VISIT BRIGHTSTORM.com FOR TONS OF VIDEO TUTORIALS AND OTHER FEATURES! http://www.brightstorm.com/ LET'S CONNECT! Facebook ► https

From playlist Geometry

👉 Learn how to define and classify different angles based on their characteristics and relationships are given a diagram. The different types of angles that we will discuss will be acute, obtuse, right, adjacent, vertical, supplementary, complementary, and linear pair. The relationships

From playlist Angle Relationships From a Figure

Complement of Independent Set is Vertex Cover | Graph Theory

We prove the complement of an independent vertex set is a vertex cover. This makes for an easy direct proof once we recall our definitions. An independent vertex set is a set of vertices, no two of which are adjacent. A vertex cover is a set of vertices such that every edge has at least on

From playlist Graph Theory

Local Statistics, Semidefinite Programming, and Community Detection - Prasad Raghavendra

Computer Science/Discrete Mathematics Seminar I Topic: Local Statistics, Semidefinite Programming, and Community Detection Speaker: Prasad Raghavendra Affiliation: University of California, Berkeley Date: May 4, 2020 For more video please visit http://video.ias.edu

From playlist Mathematics

Lecture 2: Topological Message Passing - Cristian Bodnar

Video recording of the First Italian Summer School on Geometric Deep Learning, which took place in July 2022 in Pescara. Slides: https://www.sci.unich.it/geodeep2022/slides/GDL_SummerSchool_Part1.pdf Blog post: https://towardsdatascience.com/a-new-computational-fabric-for-graph-neural-n

From playlist First Italian School on Geometric Deep Learning - Pescara 2022

Yuval Peres: Self-interacting walks and uniform spanning forests

Abstract: In the first half of the talk, I will survey results and open problems on transience of self-interacting martingales. In particular, I will describe joint works with S. Popov, P. Sousi, R. Eldan and F. Nazarov on the tradeoff between the ambient dimension and the number of differ

From playlist Probability and Statistics

Live CEOing Ep 600: Language Design Review of Graphics, Geometry and Graph Features

In this episode of Live CEOing, Stephen Wolfram discusses upcoming improvements and features to the Wolfram Language. If you'd like to contribute to the discussion in future episodes, you can participate through this YouTube channel or through the official Twitch channel of Stephen Wolfram

From playlist Behind the Scenes in Real-Life Software Design

NJIT Data Science Seminar: Steven Skiena, Stony Brook University

NJIT Institute for Data Science https://datascience.njit.edu/ Word and Graph Embeddings for Machine Learning Steven Skiena, Ph.D. Distinguished Professor Department of Computer Science Stony Brook University Distributed word embeddings (word2vec) provides a powerful way to reduce large

From playlist Talks

The abstract chromatic number - Leonardo Nagami Coregliano

Computer Science/Discrete Mathematics Seminar I Topic: The abstract chromatic number Speaker: Leonardo Nagami Coregliano Affiliation: University of Chicago Date: March 22, 2021 For more video please visit http://video.ias.edu

From playlist Mathematics

Due to the COVID-19 pandemic, Carnegie Mellon University is protecting the health and safety of its community by holding all large classes online. People from outside Carnegie Mellon University are welcome to tune in to see how the class is taught, but unfortunately Prof. Loh will not be o

From playlist CMU 21-228 Discrete Mathematics

Global symmetry from local information: The Graph Isomorphism Problem – László Babai – ICM2018

Combinatorics | Mathematical Aspects of Computer Science Invited Lecture 13.4 | 14.5 Global symmetry from local information: The Graph Isomorphism Problem László Babai Abstract: Graph Isomorphism (GI) is one of a small number of natural algorithmic problems with unsettled complexity stat

From playlist Combinatorics

Ivan Levcovitz: Right-angled Coxeter groups commensurable to right-angled Artin groups

CIRM VIRTUAL EVENT Recorded during the meeting"Virtual Geometric Group Theory conference " the May 20, 2020 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM

From playlist Virtual Conference

Identifying congruent parts between two polygons

👉 Learn how to solve with similar polygons. Two polygons are said to be similar if the corresponding angles are congruent (equal). When two polygons are similar the corresponding sides are proportional. Knowledge of the length of the sides or the proportion of the side lengths of one of th

From playlist Congruent Polygons