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A generalization of chromatic index

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. ...

##
###
Finding the chromatic number by means of critical graphs

2002
*
ACM Journal of Experimental Algorithmics
*

We propose a new exact algorithm for finding

doi:10.1145/944618.944628
fatcat:syw6sih63zbvdlh5rkpnmajz6m
*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 ...##
###
Finding the chromatic number by means of critical graphs

2000
*
Electronic Notes in Discrete Mathematics
*

We propose a new exact algorithm for finding

doi:10.1016/s1571-0653(05)80155-6
fatcat:oylwxhacqng4haq52yzkkydrn4
*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 ...##
###
Efficient algorithms for finding critical subgraphs

2008
*
Discrete Applied Mathematics
*

Computational experiments are reported

doi:10.1016/j.dam.2006.07.019
fatcat:p6dt2x6zjjegpnkhv65aynts24
*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. ...##
###
Edge chromatic 6-critical graphs

2019
*
Malaya Journal of Matematik
*

In this paper, we study

doi:10.26637/mjm0s01/0018
fatcat:e76j6uppujay3nzyuxspgtg4ru
*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*. ...##
###
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

1994
*
Graphs and Combinatorics
*

We consider

doi:10.1007/bf02986667
fatcat:7kfceirmejephkxoib4llhwb3e
*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. ...##
###
On the chromatic number of the lexicographic product and the Cartesian sum of graphs

1994
*
Discrete Mathematics
*

In this section we generalize these two results to X-

doi:10.1016/0012-365x(93)e0056-a
fatcat:553cadbqg5csvayyc4b2ht7tdu
*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. ...##
###
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

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. ...

##
###
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]

2019
*
arXiv
*
pre-print

Let n_g(k) denote

arXiv:1805.06713v4
fatcat:l6jkopozsrbnhmzfjyct4256ge
*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. ...##
###
Connected τ -critical hypergraphs of minimal size

2005
*
Discrete Mathematics & Theoretical Computer Science
*

It can be shown that a connected $τ$ -

doi:10.46298/dmtcs.3397
fatcat:xzevnh6uvzdb5jkasffq24jfqa
*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. ...
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