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Clique partitions and clique coverings

Paul Erd'́os, Ralph Faudree, Edward T. Ordman
1988 Discrete Mathematics  
times as fast as the clique covering number, where c is at least l/64.  ...  Several new tools are presented for determining the number of cliques needed to (edge-)partition a graph. For a graph on n vertices, the clique partition number can grow et-?  ...  He asks about clique coverings and clique partitions of T,,.  ... 
doi:10.1016/0012-365x(88)90197-5 fatcat:2htnjyhsgvfddlcrxrnikxq2xm

Clique covering and clique partition in generalizations of line graphs

Erich Prisner
1995 Discrete Applied Mathematics  
In this note we show that minimum sets of maximal cliques covering, respectively partitioning the edge set of a graph can be computed efficiently for certain superclasses of the class of line graphs. 0166P218X  ...  Cliques are complete subgraphs of a graph.  ...  A clique partition or maxclique partition is an edge-disjoint clique covering, respectively maxclique covering. Note that such a maxclique partition does not always exist.  ... 
doi:10.1016/0166-218x(94)00076-p fatcat:ifnamlrwazepdkngnazrreppv4

Clique coverings and partitions of line graphs

Bo-Jr Li, Gerard J. Chang
2008 Discrete Mathematics  
clique partitions and clique covers of a line graph, Discrete Math. 83 (1990) 49-62].  ...  The clique covering (partition) number cc(G) (cp(G)) of G is the minimum size of a clique covering (partition) of G.  ...  The authors thank a referee who kindly pointed out that these results are in fact old, and likes the alternative proof for the De Bruijn-Erdős Theorem.  ... 
doi:10.1016/j.disc.2007.04.059 fatcat:yeztnlgw3vhqzjoltbqdk6luty

On the number of distinct minimal clique partitions and clique covers of a line graph

Sean McGuinness, Rolf Rees
1990 Discrete Mathematics  
A minimal clique covering (partition) P is a clique covering (partition) with IPI = cc(G) (IPI = cp(G)).  ...  The clique covering number cc(G) is the quantity min{ IPI : P is a clique covering of G}; the clique partition number cp(G) is the quantity min{lPl : P is a clique partition of G}.  ... 
doi:10.1016/0012-365x(90)90220-c fatcat:r2rdf3haznga5p24rz32jfvtpe

Maximal-clique partitions of interval graphs

Ma Shaohan, W. D. Wallis
1988 Journal of the Australian Mathematical Society  
It is shown that if an interval graph possesses a maximal-clique partition then its clique covering and clique partition numbers are equal, and equal to the maximal-clique partition number.  ...  Moreover an interval graph has such a partition if and only if all its maximal cliques are edge-disjoint . 1980 Mathematics subject classification (Amer. Math. Soc): 05 C 35.  ...  If the clique covering C has cardinality \C\ and |CX'| > \C\ for all clique coverings C' of G, then C is called a minimal clique covering and the clique covering number of G, cc(G), is defined to equal  ... 
doi:10.1017/s1446788700030147 fatcat:nrnxawvl6vakzdrybvfhroivye

Complexity results on graphs with few cliques

Bill Rosgen, Lorna Stewart
2007 Discrete Mathematics & Theoretical Computer Science  
Several classes of graphs which have few cliques are discussed, and the complexity of some partitioning and covering problems are determined for the class of all graphs which have fewer cliques than a  ...  Other problems, such as the vertex clique cover and edge clique cover problems remain NP-complete on these classes.  ...  Acknowledgements This research was partially supported by NSERC and iCORE.  ... 
doi:10.46298/dmtcs.387 fatcat:zd566nxpufhvrfaej6urs5l7nm

Clique Partitions of Glued Graphs

Chariya Uiyyasathian, Uthoomporn Jongthawonwuth
2010 Journal of Mathematics Research  
Let P and P be minimum clique partitions of G and G − C, respectively. Then P ∪ {C} is a clique partition of G, so cp(G) ≤ |P ∪ {C}| and hence cp(G) − 1 ≤ cp(G − C).  ...  For any glued graphs of G 1 and G 2 at a clone H, since the union of a minimum clique partition of G 1 (or G 2 ) and minimum clique partition of G 2 − H (or G 1 − H) provides a clique partition of G 1  ...  The clique covering number cc (G) and clique partition number cp(G) are the smallest cardinality among all clique coverings and clique partitions of G, respectively.  ... 
doi:10.5539/jmr.v2n2p104 fatcat:4pq4ibdbqfcjvpm3ixi2cywouu

Clique coverings of graphs V: maximal-clique partitions

N.J. Pullman, H. Shank, W.D. Wallis
1982 Bulletin of the Australian Mathematical Society  
the maximal-clique partition number and its relationship to other clique covering and partition numbers.  ...  I is |C'| t |C| for all clique coverings C' of G , then C is called a minimum clique covering and |C| is called the clique covering number of G , denoted by cc(G) .  ...  The clique A ' must be in M because M covers H and no member of B covers A ' . Now M' = (MV4') u {A} is a maximal-clique partition of G and hence is either A or B . But A If.  ... 
doi:10.1017/s0004972700005414 fatcat:z4b6ulxa5rhvrgdtapbelizgu4

On the minimum monochromatic or multicolored subgraph partition problems

Xueliang Li, Xiaoyan Zhang
2007 Theoretical Computer Science  
We investigate the complexity of the problems for finding the minimum number of monochromatic or multicolored subgraphs, such as cliques, cycles, trees and paths, partitioning V (G), depending on the number  ...  We also present a greedy scheme that yields a (ln m + 1)-approximation for the problem of finding the minimum number of monochromatic cliques partitioning V (G) for a K − 4 -free graph G, where m is the  ...  Baogang Xu and Prof. Shenggui Zhang for helpful discussions and the referees for their helpful suggestions.  ... 
doi:10.1016/j.tcs.2007.04.033 fatcat:6ny45ast65c4bmtbal3nxxbkk4

Clique Cover Width and Clique Sum [article]

Farhad Shahrokhi
2015 arXiv   pre-print
When G is the clique sum of G_1 and G_2, we prove that CCW(G) < 3/2(CCW(G_1) + CCW(G_2)).  ...  The clique cover width of G, denoted by CCW(G), is the minimum value of the bandwidth of all clique cover graphs of G.  ...  ., q, we define the distance of strips S i and S j , in the partition P (C, B), to be |j − i|. Proposition 2.1 Let C be a clique cover in G and let B be a block in C.  ... 
arXiv:1502.06165v2 fatcat:raa66usvvvbirpwxt2p3kyfjaq

Recent examples in the theory of partition graphs

D.W. DeTemple, M.J. Dineen, J.M. Robertson, K.L. McAvaney
1993 Discrete Mathematics  
Robertson and K.L.  ...  Two questions which arose in the study of partition graphs are answered by recently discovered examples. An enumeration of the partition graphs on ten or fewer vertices is provided.  ...  A clique cover is minimal if no proper subcollection of its cliques is also a clique cover. If a clique cover which satisfies Condition I exists, then it can be taken to be minimal.  ... 
doi:10.1016/0012-365x(93)90520-4 fatcat:spgwc2hn4bh35lzpbjfpua4u7e

Sigma clique covering of graphs [article]

Akbar Davoodi, Ramin Javadi, Behnaz Omoomi
2015 arXiv   pre-print
G, covering (resp. partitioning) all edges of G such that the sum of sizes of the cliques is at most k.  ...  The sigma clique cover number (resp. sigma clique partition number) of graph G, denoted by scc(G) (resp. scp(G)), is defined as the smallest integer k for which there exists a collection of cliques of  ...  The minimum size of a clique covering, a biclique covering, a clique partition and a biclique partition of G are called clique cover number, biclique cover number, clique partition number and biclique  ... 
arXiv:1503.02380v1 fatcat:2vewghybarhgnefn2zuld5gnae

On Clique Coverings of Complete Multipartite Graphs [article]

Akbar Davoodi, Dániel Gerbner, Abhishek Methuku, Máté Vizer
2018 arXiv   pre-print
A clique covering of a graph G is a set of cliques of G such that any edge of G is contained in one of these cliques, and the weight of a clique covering is the sum of the sizes of the cliques in it.  ...  The sigma clique cover number scc(G) of a graph G, is defined as the smallest possible weight of a clique covering of G. Let K_t(d) denote the complete t -partite graph with each part of size d.  ...  Research of Gerbner and Methuku was supported by the National Research, Development and Innovation Office -NKFIH, grant K 116769.  ... 
arXiv:1809.01443v1 fatcat:oxpa2njnubcytaxbnbxw34bjmi

On a k-clique-join of a class of partitionable graphs

Mihai Talmaciu
2005 Computer Science Journal of Moldova  
We call a graph G O-graph if there is an optimal coloring of the set of vertices and an optimal (disjoint) covering with cliques such that any class of colors intersects any clique.  ...  In this paper, it has been established the relation to [p,q,r]-partite graphs and the fact that the O-graphs admit a k-clique-join. American Mathematical Society (2000): 05C17.  ...  Let G = (V, E) be an O-graph and (S 1 , ..., S ω ) a partition in ω α-stable sets of G and (Q 1 , ..., Q α ) a disjoint covering with α ω-cliques of begin H : H =G; while (∃e = xy ∈ E with {x, y} ⊂ Q  ... 
doaj:2fa135775a1b424085de547a36172c7f fatcat:hkazvssbyvdntg7yiecoyhm2la

Clique Cover on Sparse Networks [chapter]

Mathieu Blanchette, Ethan Kim, Adrian Vetta
2012 2012 Proceedings of the Fourteenth Workshop on Algorithm Engineering and Experiments (ALENEX)  
Empirical studies show that our algorithms are both efficient and practical on actual simulated and biological networks, and that the clique covers obtained on real networks yield biological insights.  ...  We consider the problem of edge clique cover on sparse networks and study an application to the identification of overlapping protein complexes for a network of binary protein-protein interactions.  ...  Therefore, we can partition the edges in E(I ∪ P ) into F 1 and F 2 , where F 1 is to be covered by cliques in X 1 , and F 2 is covered by cliques in X 2 .  ... 
doi:10.1137/1.9781611972924.10 dblp:conf/alenex/BlanchetteKV12 fatcat:oo2qzwd7s5c57nst5kgn4upxti
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