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A generalization of chromatic index

E. Sampathkumar, G.D. Kamath
1994 Discrete Mathematics  
The general chromatic index x;(G) of G is the minimum order of a partition P of E such that for any set F in P every component in the subgraph (F) induced by F has size at most k-1.  ...  This paper initiates a study of x;(G) and generalizes some known results on chromatic index.  ...  Some elementary properties of (k, n)-critical graphs are as follows. every component in the subgraph (F) induced by F has size at most k -1.  ... 
doi:10.1016/0012-365x(94)90088-4 fatcat:iukdqssjjreatofdxmjxg2qjli

Finding the chromatic number by means of critical graphs

Francine Herrmann, Alain Hertz
2002 ACM Journal of Experimental Algorithmics  
We propose a new exact algorithm for finding the chromatic number of a graph G.  ...  Such a subgraph is said critical since all proper induced sub-graphs of G have a chromatic number strictly smaller than G .  ...  of number of number of chromatic size of backtracks backtracks graph vertices edges number critical subgraph New EXACT anna 138 493 11 11 1 8 david 87 406 11 11 1 1 huck 74  ... 
doi:10.1145/944618.944628 fatcat:syw6sih63zbvdlh5rkpnmajz6m

Finding the chromatic number by means of critical graphs

Francine Herrmann, Alain Hertz
2000 Electronic Notes in Discrete Mathematics  
We propose a new exact algorithm for finding the chromatic number of a graph G.  ...  Such a subgraph is said critical since all proper induced sub-graphs of G have a chromatic number strictly smaller than G .  ...  of number of number of chromatic size of backtracks backtracks graph vertices edges number critical subgraph New EXACT anna 138 493 11 11 1 8 david 87 406 11 11 1 1 huck 74  ... 
doi:10.1016/s1571-0653(05)80155-6 fatcat:oylwxhacqng4haq52yzkkydrn4

Efficient algorithms for finding critical subgraphs

C. Desrosiers, P. Galinier, A. Hertz
2008 Discrete Applied Mathematics  
Computational experiments are reported on random and DIMACS benchmark graphs to compare the proposed algorithms, as well as to find lower bounds on the chromatic number of these graphs.  ...  This paper presents algorithms to find vertex-critical and edge-critical subgraphs in a given graph G, and demonstrates how these critical subgraphs can be used to determine the chromatic number of G.  ...  A lower bound on the size of minimum critical subgraphs In some cases, it can be useful to have an indication on the size of minimum critical subgraphs.  ... 
doi:10.1016/j.dam.2006.07.019 fatcat:p6dt2x6zjjegpnkhv65aynts24

Edge chromatic 6-critical graphs

J. Sakila Devi
2019 Malaya Journal of Matematik  
In this paper, we study the structure of edge chromatic 6-critical graphs of three different classes by obtaining the bounds on the number of major vertices and the size.  ...  This paper gives new bounds on the number of major vertices and the size of three different classes of edge chromatic 6-critical graphs.  ...  Also improvement on Fiorini's bound on the number of major vertices is obtained in all the three classes of edge chromatic 6-critical graphs.  ... 
doi:10.26637/mjm0s01/0018 fatcat:e76j6uppujay3nzyuxspgtg4ru

Page 4625 of Mathematical Reviews Vol. , Issue 2000g [page]

2000 Mathematical Reviews  
We obtain a new upper bound for the harmonious chromatic number of general graphs.” 2000g:05064 05C15 Haile, Dawit Bounds on the size of critical edge-chromatic graphs.  ...  We give new lower bounds for the size of A-critical edge-chromatic graphs, for A>9” 2000g:05065 05C15 0s5C05 Jiang, Tao [Jiang, Tao*] (1-IL; Urbana, IL); West, Douglas B. (1-IL; Urbana, IL) Coloring of  ... 

Page 4149 of Mathematical Reviews Vol. , Issue 93h [page]

1993 Mathematical Reviews  
A connected graph G is called A-critical if A(G) = A, x'(G) = A+1 and a removal of any edge of G decreases the chromatic index.  ...  For k = 3, the k-cyclic chromatic number coincides with the chromatic number of the graph. For k = 4, it is at most 6 and this upper bound is tight.  ... 

Size, chromatic number, and connectivity

J. Bhasker, Tariq Samad, Douglas B. West
1994 Graphs and Combinatorics  
We consider the minimum number of edges in a k-edge-connected graph of order n with chromatic number at least c, obtaining the optimal bounds in most cases.  ...  We are greatly indebted to an anonymous referee, whose comments led to much stronger results than were present in the paper when originally submitted.  ...  Since a c-critical graph is c -1-edge-connected, we may assume k = c -1, since the desired bound in this case is larger than the desired bound for smaller k.  ... 
doi:10.1007/bf02986667 fatcat:7kfceirmejephkxoib4llhwb3e

On the chromatic number of the lexicographic product and the Cartesian sum of graphs

Niko Čižek, Sandi Klavžar
1994 Discrete Mathematics  
In this section we generalize these two results to X-critical graphs. But first we summarise some basic observations on chromatic numbers of the Cartesian sum of graphs. Proposition 4.  ...  Geller and Stahl [3] proved that if G has at least one edge, then z(G[H])>~x(G)+2)~(H)--2 for any graph H. This bound is the best general lower bound known so far.  ... 
doi:10.1016/0012-365x(93)e0056-a fatcat:553cadbqg5csvayyc4b2ht7tdu

Page 6613 of Mathematical Reviews Vol. , Issue 99j [page]

1999 Mathematical Reviews  
Hajos proved that every k-critical graph is Hajos-k-constructible and asked the question whether one can obtain every k-critical graph in such a way that all intermediary graphs are also k-critical.  ...  (F-PARIS6-CM; Paris) A bound on the total chromatic number. (English summary) Combinatorica 18 (1998), no. 2, 241-280.  ... 

Page 1425 of Mathematical Reviews Vol. 51, Issue 5 [page]

1976 Mathematical Reviews  
The paper describes several general construc- tions to (a) obtain critical graphs from critical hypergraphs, (b) increase the size of edges in critical hypergraphs, (c) decrease the size of edges to 3  ...  The paper concludes with rather sharp upper and lower bounds on the maximum number of edges in critical k-chromatic hypergraphs. {For the entire collection see MR 50 #12526.}  ... 

The b-chromatic number and related topics—A survey

Marko Jakovac, Iztok Peterin
2018 Discrete Applied Mathematics  
The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-coloring in which every color class contains at least one vertex that has a neighbor in each of the other color  ...  In this survey we present the most important results on b-colorings, b-chromatic number and related topics.  ...  b-critical graphs A graph G is edge b-critical if the removal of any edge of G decreases its b-chromatic number.  ... 
doi:10.1016/j.dam.2017.08.008 fatcat:cn2yvemuszgu3nfel43mlnbnru

Page 54 of Mathematical Reviews Vol. , Issue 2002A [page]

2002 Mathematical Reviews  
(D-ILTL On the number of edges in colour-critical graphs and hypergraphs.  ...  2002a:05098 of G, called the cyclic chromatic number of G. Plummer and Toft conjectured that y.(G) < A*(G) +2, where A*(G) is the maximum face size of G.  ... 

Bounds for the smallest k-chromatic graphs of given girth [article]

Geoffrey Exoo, Jan Goedgebeur
2019 arXiv   pre-print
Let n_g(k) denote the smallest order of a k-chromatic graph of girth at least g. We consider the problem of determining n_g(k) for small values of k and g.  ...  After giving an overview of what is known about n_g(k), we provide some new lower bounds based on exhaustive searches, and then obtain several new upper bounds using computer algorithms for the construction  ...  Most of the computations were carried out using the Stevin Supercomputer Infrastructure at Ghent University and the computers at the Computer Science labs at Indiana State University.  ... 
arXiv:1805.06713v4 fatcat:l6jkopozsrbnhmzfjyct4256ge

Connected τ -critical hypergraphs of minimal size

Matěj Stehlík
2005 Discrete Mathematics & Theoretical Computer Science  
It can be shown that a connected $τ$ -critical hypergraph $\mathscr{H}$ has at least $2τ (\mathscr{H})-1$ edges; this generalises a classical theorem of Gallai on $χ$ -vertex-critical graphs with connected  ...  Finally, we deduce some new results on $χ$ -vertex-critical graphs with connected complements.  ...  We first present a sharp lower bound on the number of edges in a connected τ -critical hypergraph, and then investigate the cases where equality is attained.  ... 
doi:10.46298/dmtcs.3397 fatcat:xzevnh6uvzdb5jkasffq24jfqa
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