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complete bipartite graph
List Colourings of Complete Multipartite Graphs with 2 Big Parts ★★
Author(s): Allagan
Question Given
, what is the smallest integer
such that
?
![$ a,b\geq2 $](/files/tex/7ae98b0c55d36dcaf7def20925869c5d4b1d48ef.png)
![$ t\geq0 $](/files/tex/13168e3697d72526aed2db7c8042d868fd46bc7e.png)
![$ \chi_\ell(K_{a,b}+K_t)= \chi(K_{a,b}+K_t) $](/files/tex/bd7ca9e1076d95097ac657d98e06895400a68ec0.png)
Keywords: complete bipartite graph; complete multipartite graph; list coloring
Alon-Saks-Seymour Conjecture ★★★
Author(s): Alon; Saks; Seymour
Conjecture If
is a simple graph which can be written as an union of
edge-disjoint complete bipartite graphs, then
.
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ m $](/files/tex/ddaab6dc091926fb1da549195000491cefae85c1.png)
![$ \chi(G) \le m+1 $](/files/tex/0aefe0f752ac272776eec5c61c7d1a1822a3d224.png)
Keywords: coloring; complete bipartite graph; eigenvalues; interlacing
The Crossing Number of the Complete Bipartite Graph ★★★
Author(s): Turan
The crossing number of
is the minimum number of crossings in all drawings of
in the plane.
Conjecture
![$ \displaystyle cr(K_{m,n}) = \floor{\frac m2} \floor{\frac {m-1}2} \floor{\frac n2} \floor{\frac {n-1}2} $](/files/tex/aecee36502739c16e07eea1fc64a43a9e95e9374.png)
Keywords: complete bipartite graph; crossing number
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