Graph Theory » Coloring

Total Colouring Conjecture ★★★

Author(s): Behzad

Conjecture   A total coloring of a graph $ G = (V,E) $ is an assignment of colors to the vertices and the edges of $ G $ such that every pair of adjacent vertices, every pair of adjacent edges and every vertex and incident edge pair, receive different colors. The total chromatic number of a graph $ G $, $ \chi''(G) $, equals the minimum number of colors needed in a total coloring of $ G $. It is an old conjecture of Behzad that for every graph $ G $, the total chromatic number equals the maximum degree of a vertex in $ G $, $ \Delta(G) $ plus one or two. In other words, \[\chi''(G)=\Delta(G)+1\ \ or \ \ \Delta(G)+2.\]

Keywords: Total coloring

Posted by Iradmusa
updated June 4th, 2008
add new comment
Graph Theory » Coloring » Vertex coloring

Hedetniemi's Conjecture ★★★

Author(s): Hedetniemi

Conjecture   If $ G,H $ are simple finite graphs, then $ \chi(G \times H) = \min \{ \chi(G), \chi(H) \} $.

Here $ G \times H $ is the tensor product (also called the direct or categorical product) of $ G $ and $ H $.

Keywords: categorical product; coloring; homomorphism; tensor product

Posted by mdevos
updated March 1st, 2020
5 comments
Graph Theory

Edge Reconstruction Conjecture ★★★

Author(s): Harary

Conjecture  

Every simple graph with at least 4 edges is reconstructible from it's edge deleted subgraphs

Keywords: reconstruction

Posted by melch
updated May 23rd, 2008
add new comment