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

Edita Máčajová, Ján Mazák
2011 Lecture Notes in Computer Science
In [12], we have shown that the acyclic 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.  ...

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?  ...

Local chromatic number and topology

Gábor Simonyi, Gábor Tardos
2005 Discrete Mathematics & Theoretical Computer Science
We further elaborate on the case of \$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.  ...

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.  ...

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  ...

Coloring Eulerian triangulations of the projective plane

Bojan Mohar
2002 Discrete Mathematics
A simple characterization of the 3, 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.  ...

Carsten Thomassen
2004 Journal of combinatorial theory. Series B (Print)
Erd + os asked if the removal of few edges in a large 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.  ...

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.  ...

Hadwiger conjecture for 8-coloring graph [article]

T.-Q. Wang, X.-J. Wang
2021 arXiv   pre-print
Therefore, we put forward a brand new 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.  ...

On acyclic colorings of graphs on surfaces

Noga Alon, Bojan Mohar, Daniel P. Sanders
1996 Israel Journal of Mathematics
This paper shows that the acyclic 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.  ...

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).  ...

Adaptable chromatic number of graph products

Pavol Hell, Zhishi Pan, Tsai-Lien Wong, Xuding Zhu
2009 Discrete Mathematics
The adaptable 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.  ...

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  ...

On building 4-critical plane and projective plane multiwheels from odd wheels [article]

Dainis Zeps
2012 arXiv   pre-print
We build unbounded classes of plane and 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.  ...

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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