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An Algorithm for Optimal Acyclic Edge-Colouring of Cubic Graphs
[chapter]

2011
*
Lecture Notes in Computer Science
*

In [12], we have shown that the acyclic

doi:10.1007/978-3-642-21204-8_17
fatcat:bxdyxkl3ivetvm4ilkpc2tvbbu
*chromatic*index of a connected subcubic*graph*G is at most*4*with the exception of K4 and K3,3, for which five colors are optimal. ... The acyclic*chromatic*index of a*graph*G is the smallest possible number of colours in an acyclic edge-colouring of G. ... Since the acyclic*chromatic*index of the*graphs*K*4*and K 3,3 is 5, there is no such algorithm for all cubic*graphs*. ...##
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Page 1010 of Mathematical Reviews Vol. 20, Issue 9
[page]

1959
*
Mathematical Reviews
*

This is an improvement on the theorem of the reviewer that every critical

*4*-*chromatic**graph*contains as a sub-*graph*a complete*4*-*graph*or a subdivision of a complete*4*-*graph*. ... Of node-critical*4*-*chromatic**graphs*with »=—6m-+-*4*nodes, for all m21, in which the number of edges is (n? ...##
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Local chromatic number and topology

2005
*
Discrete Mathematics & Theoretical Computer Science
*

We further elaborate on the case of $

doi:10.46298/dmtcs.3447
fatcat:dnpzcmfgdffnpp3rduueiuny4i
*4*$-*chromatic**graphs*and, in particular, on surface quadrangulations. ... In particular, we determine the local*chromatic*number of certain odd*chromatic*Schrijver*graphs*and generalized Mycielski*graphs*. ... It is known that generalized Mycielski*graphs*of*chromatic*number*4*quadrangulate the*projective*plane. ...##
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Page 5776 of Mathematical Reviews Vol. , Issue 95j
[page]

1995
*
Mathematical Reviews
*

Moreover, it is shown that the result is best possible, in the sense that there are infinitely many toroidal, triangle-free,

*4*-*chromatic**graphs*. ... White (1-WMI; Kalamazoo, MI) 95j:05099 05C15 05C10 Zaslavsky, Thomas (1-SUNY2; Binghamton, NY) The signed*chromatic*number of the*projective*plane and Klein bottle and antipodal*graph*coloring. ...##
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Page 1891 of Mathematical Reviews Vol. , Issue 89D
[page]

1989
*
Mathematical Reviews
*

Consistently there exist X2-

*chromatic**graphs*with no X,-*chromatic*subgraphs. ... The statement that every uncountably*chromatic**graph*of size 8, contains an uncountably*chromatic*w-connected sub-*graph*is consistent and independent. It is consistent that there is ...##
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Coloring Eulerian triangulations of the projective plane

2002
*
Discrete Mathematics
*

A simple characterization of the 3,

doi:10.1016/s0012-365x(01)00092-9
fatcat:3k5ym5w6lbatziaw2gyt7h3y4q
*4*, or 5-colorable Eulerian triangulations of the*projective*plane is given. ... It is easy to find examples on the*projective*plane whose*chromatic*number is equal to 3,*4*, or 5, respectively, and it is easy to see that the*chromatic*number of an Eulerian triangulation of the*projective*... Youngs [7] proved that a quadrangulation Q of the*projective*plane which is not 2-colorable is neither 3-colorable, and its*chromatic*number is*4*. ...##
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Quadrangulations and 4-color-critical graphs

2004
*
Journal of combinatorial theory. Series B (Print)
*

Erd + os asked if the removal of few edges in a large

doi:10.1016/j.jctb.2003.11.003
fatcat:rz2cuhkbjnhujcpqnsg4odvmp4
*4*-color-critical*graph*always leaves a 3-*chromatic**graph*. ... We answer both problems, which are stated in Bollobas' monograph Extremal*Graph*Theory from 1978, in the negative. r ... Introduction How close can a*4*-*chromatic**graph*be to a bipartite*graph*? Bolloba´s' monograph [2] contains two problems on that question. ...##
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Page 2465 of Mathematical Reviews Vol. , Issue 89E
[page]

1989
*
Mathematical Reviews
*

G. (

*4*-LANC); Hilton, A. J. W. (*4*-LANC) The existence of multigraphs with a given degree sequence and given*chromatic*index. ... Andrew Vince (1-FL) 89e:05086 05C15 Bollobas, B. (*4*-CAMB); Harris, A. J. (*4*-CAMB) List-colourings of*graphs*.*Graphs*Combin. 1 (1985), no. 2, 115-127. ...##
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Hadwiger conjecture for 8-coloring graph
[article]

2021
*
arXiv
*
pre-print

Therefore, we put forward a brand new

arXiv:2104.13519v1
fatcat:l54jt6kdrrbelhkivblfosxtai
*chromatic**graph*configuration and show how to describe the*graph*coloring issues in*chromatic*space. ... A few cases of Hadwiger Conjecture, such as 1, 2, 3,*4*, 5, 6-colorable*graphs*have been completely proved to convince all1-5, but the proofs are tremendously difficult for over the 5-colorable graph6,7 ... On another hand, if we know the*graph*with minor K9, then we can assign the*graph*to two CHPs, which the first CHP is with*chromatic*number*4*. The second CHP is with*chromatic*number at most*4*. ...##
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On acyclic colorings of graphs on surfaces

1996
*
Israel Journal of Mathematics
*

This paper shows that the acyclic

doi:10.1007/bf02762708
fatcat:joib2strubcwfetezhruoli5p4
*chromatic*number of the*projective*plane is at most 7. The acyclic*chromatic*number of an arbitrary surface with Euler characteristic χ = −γ is at most O(γ*4*/7 ). ... This is nearly tight; for every γ > 0 there are*graphs*embeddable on surfaces of Euler characteristic −γ whose acyclic*chromatic*number is at least Ω(γ*4*/7 /(log γ) 1/7 ). ...*Projective*Plane*Graphs*This section presents a 7-color theorem for*projective*planar*graphs*. This result may not be best possible. ...##
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Page 1394 of Mathematical Reviews Vol. , Issue 98C
[page]

1998
*
Mathematical Reviews
*

The d-diagonal

*chromatic*number of 3-connected plane*graphs*is at least A(A — 1)*4*/? for even d and at least 1 + 3(A—2)(A—1)'*4*-)/? for d odd. Theorem 5. ... Let P(G,A) be the*chromatic*polynomial of a*graph*G. Welsh and Brenti made the following Conjecture |: For all integers*4*> 1, P?(G,*4*) > P(G,A—1)P(G,A+1). ...##
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Adaptable chromatic number of graph products

2009
*
Discrete Mathematics
*

The adaptable

doi:10.1016/j.disc.2009.05.029
fatcat:vidzhpzclfcmlolg6w5jfq35me
*chromatic*number of a*graph*G is smaller than or equal to the ordinary*chromatic*number of G. ... When G is complete, we prove this conjecture with k ≥*4*, and offer additional evidence suggesting it may hold with k ≥ 2. We also discuss other products and propose several open problems. ... So φ(K*4*n , n) ≥ n 3 (n − 1) 3 ≥ (n − 1) 2 (n*4*ln 2 + ln(n − 1)) (for n ≥*4*). Squares of K n Theorem*4*shows that K*4*n has its adaptable*chromatic*number equal to its*chromatic*number. ...##
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Page 45 of Mathematical Reviews Vol. , Issue 93a
[page]

1993
*
Mathematical Reviews
*

Albertson (1-SMTH)
93a:05063 05C15 05C70 Wan, Hong Hui (PRC-HUST); Cheng, Zai Shu (PRC-HUST) A note on the

*graphs*of class one. Math. Appl.*4*(1991), no.*4*, 128-129. ... Summary: “In this paper, the concept of*chromatic*decomposition of triangulated*graphs*is introduced and the relations between the coefficients of the*chromatic*decomposition of a triangulated*graph*93a ...##
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On building 4-critical plane and projective plane multiwheels from odd wheels
[article]

2012
*
arXiv
*
pre-print

We build unbounded classes of plane and

arXiv:1202.4862v1
fatcat:24iipbchb5dkxnx62vd4lj4ux4
*projective*plane multiwheels that are*4*-critical that are received summing odd wheels as edge sums modulo two. ... All*graphs*of these classes belong to 2n-2-edges-class of*graphs*, among which are those that quadrangulate*projective*plane, i.e.,*graphs*from Gr\"otzsch class, received applying Mycielski's Construction ... The resulting*graph*is 3-*chromatic*. We need one more crucial feature of*4*-critical*graphs*. ...##
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Algorithms
[chapter]

2011
*
Graph Coloring Problems
*

*Graphs*Without Odd-K 5 115 6.6 Scheme Conjecture 115 6.7

*Chromatic*

*4*-Schemes 116 6.8 Odd Subdivisions of K

*4*116 6.9 Nonseparating Odd Cycles in

*4*-Critical

*Graphs*117 6.10 Minimal Edge ... Erdos' Property B Property B(s) Finite

*Projective*Planes Steiner Triple Systems Steiner Quadruple Systems._ Minimum-Weight 3-

*Chromatic*Hypergraphs Positional Games Tic-Tac-Toe Square Hypergraphs Size ...

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