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### Bounds for mean colour numbers of graphs

F.M. Dong
2003 Journal of combinatorial theory. Series B (Print)
This result gives a method to find upper bounds and lower bounds for the mean colour number of any graph. We also prove that mðGÞomðG,K 1 Þ for an arbitrary graph G: r  ...  Let mðGÞ denote the mean colour number of a graph G: Mosca discovered some counterexamples which disproved a conjecture proposed by Bartels and Welsh that if H is a subgraph of G; then mðHÞpmðGÞ: In this  ...  Jim Geelen for his suggestion to study upper bounds for mean colour numbers by tree widths, and also thanks the referees for their valuable comments.  ...

### A Note on Lower Bounds for Induced Ramsey Numbers [article]

Izolda Gorgol
2017 arXiv   pre-print
Firstly there will be shown that the lower bound of the induced Ramsey number for a connected graph G with independence number α and a graph H with clique number ω roughly ω^2α/2.  ...  We say that a graph F strongly arrows a pair of graphs (G,H) if any 2-colouring of its edges with red and blue leads to either a red G or a blue H appearing as induced subgraphs of F.  ...  To prove the lower bound for the induced Ramsey number we should show that we can colour every graph F with prescribed number of vertices without induced monochromatic copies of given graphs (G, H).  ...

### Ramsey numbers in complete balanced multipartite graphs. Part I: Set numbers

Alewyn P. Burger, Jan H. van Vuuren
2004 Discrete Mathematics
The notion of a graph theoretic Ramsey number is generalised by assuming that both the original graph whose edges are arbitrarily bi-coloured and the sought after monochromatic subgraphs are complete,  ...  In this paper the deÿnition of a multipartite Ramsey number is broadened still further, by incorporating o -diagonal numbers, ÿxing the number of vertices per partite set in the larger graph and then seeking  ...  of Stellenbosch.  ...

### Ramsey numbers in complete balanced multipartite graphs. Part I: Set numbers

A BURGER
2004 Discrete Mathematics
The notion of a graph theoretic Ramsey number is generalised by assuming that both the original graph whose edges are arbitrarily bi-coloured and the sought after monochromatic subgraphs are complete,  ...  In this paper the deÿnition of a multipartite Ramsey number is broadened still further, by incorporating o -diagonal numbers, ÿxing the number of vertices per partite set in the larger graph and then seeking  ...  of Stellenbosch.  ...

### Hat guessing numbers of strongly degenerate graphs [article]

Charlotte Knierim, Anders Martinsson, Raphael Steiner
2021 arXiv   pre-print
any class of K_2,s-free graphs with bounded expansion, such as the class of C_4-free planar graphs, more generally K_2,s-free graphs with bounded Hadwiger number or without a K_t-subdivision, and for  ...  As a consequence, we significantly improve the best known upper bound on the hat guessing number of outerplanar graphs from 2^125000 to 40, and further derive upper bounds on the hat guessing number for  ...  More bounds on hat guessing numbers for specific graph classes can be found in [4, 12, 13].  ...

### On the Delta(d)–chromatic number of a complete balanced multipartite graph

AP Burger, I Nieuwoudt, JH Van Vuuren
2007 ORiON
Dedicated with deep respect and appreciation to our friend, teacher and mentor, Gerhard Geldenhuys, who introduced us to the sheer beauty of graph theory in the early nineteen nineties, and who fostered  ...  in us an affinity for hard operations research.  ...  Work towards this paper was supported financially by Research Subcommittee B of the University of Stellenbosch in the form of a post-doctoral fellowship for the first author and by the South African National  ...

### Enumerating maximal independent sets with applications to graph colouring

Jesper Makholm Byskov
2004 Operations Research Letters
As an application of the proof, we construct improved algorithms for graph colouring and computing the chromatic number of a graph.  ...  We give tight upper bounds on the number of maximal independent sets of size k (and at least k and at most k) in graphs with n vertices.  ...  ., o(2 n ) upper bound on the number of maximal 3-colourable subgraphs in a graph.  ...

### Chromatic Numbers of Exact Distance Graphs [article]

Jan van den Heuvel, H.A. Kierstead, Daniel A. Quiroz
2018 arXiv   pre-print
For odd p, the existing lower bound on the number of colours needed to colour G^[ p] when G is planar is improved. Similar lower bounds are given for K_t-minor free graphs.  ...  In particular, we show that for any graph G and odd positive integer p, the chromatic number of G^[ p] is bounded by the weak (2p-1)-colouring number of G.  ...  Acknowledgement The authors would like to thank the anonymous referees for careful reading and for their corrections and suggestions.  ...

### Improvements to MCS algorithm for the maximum clique problem

Mikhail Batsyn, Boris Goldengorin, Evgeny Maslov, Panos M. Pardalos
2013 Journal of combinatorial optimization
In this paper we present improvements to one of the most recent and fastest branch-and-bound algorithm for the maximum clique problem-MCS algorithm by Tomita et al.  ...  The suggested improvements include: incorporating of an efficient heuristic returning a high-quality initial solution, fast detection of clique vertices in a set of candidates, better initial colouring  ...  Acknowledgements The authors would like to thank professor Mauricio Resende and his co-authors for the source code of their powerful ILS heuristic.  ...

### Ramsey Numbers of Squares of Paths

Peter Allen, Barnaby Roberts, Jozef Skokan
2015 Electronic Notes in Discrete Mathematics
The Ramsey number R(G, H) has been actively studied for the past 40 years, and it was determined for a large family of pairs (G, H) of graphs.  ...  The Ramsey number of paths was determined very early on, but surprisingly very little is known about the Ramsey number for the powers of paths.  ...  Introduction For graphs G and H, the Ramsey number R(G, H) is the least natural number N such that any colouring of the edges of the complete graph K N on N vertices with blue and red contains either a  ...

### On the mean chromatic number

Martin Anthony
1994 Discrete Mathematics
The mean chromatic number of a graph is a measure of the expected performance of the greedy vertex-colouring algorithm when each ordering of the vertices is equally likely.  ...  Some results on the value of the mean chromatic number and its asymptotic behaviour are presented.  ...  The mean chromatic number Taking the colours to be the positive integers, the greedy vertex-colouring algorithm can be described as follows.  ...

### Colouring Non-sparse Random Intersection Graphs [chapter]

Sotiris Nikoletseas, Christoforos Raptopoulos, Paul G. Spirakis
2009 Lecture Notes in Computer Science
This means that even for quite dense graphs, using the same number of colours as those needed to properly colour the clique induced by any label suffices to colour almost all of the vertices of the graph  ...  We study here the problem of efficiently coloring (and of finding upper bounds to the chromatic number) of RIGs.  ...  This means that even for quite dense graphs, using the same number of colours as those needed to properly colour the clique induced by any label suffices to colour almost all of the vertices of the graph  ...

### Recent results of Novosibirsk mathematicians in graph theory

L. S. Mel'nikov
1987 Časopis pro pěstování matematiky
Ringel  obtained such an upper bound of the chromatic number ^(N) g [(9 + 7(81 -32N))/2] for N ^ 2.  ...  chromatic number of graphs admitting such a 1-embedding.  ...

### Bipartite Ramsey numbers and Zarankiewicz numbers

Wayne Goddard, Michael A. Henning, Ortrud R. Oellermann
2000 Discrete Mathematics
In this paper we calculate small exact values of z(s, 2) and determine bounds for Zarankiewicz numbers in general. The latter are used to bound b(m, n) for m, n ≤ 6.  ...  The bipartite Ramsey number b(m, n) is the least positive integer b such that if the edges of K(b, b) are coloured with red and blue, then there always exists a blue K(m, m) or a red K(n, n).  ...  In particular, one looks for an upper bound for the number of red edges and an upper bound for the number of blue edges so that the total number of edges exceeds the sum of these bounds.  ...

### The Proportional Colouring Problem: Optimizing Buffers in Wireless Mesh Networks

Florian Huc, Cláudia Linhares-Sales, Hervé Rivano
2008 Electronic Notes in Discrete Mathematics
We also give a lower and an upper bound that can be polynomially computed. We finally characterize some graphs and weighted graphs for which we can determine the proportional edge chromatic number.  ...  If such colouring exists, we want to find one using the minimum number of colours.  ...  By "proper", we mean that, for any two adjacent edges, the sets of assigned colours have an empty intersection.  ...
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