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If G is a graph on n vertices and r 2 2, w e let m,(G) denote the minimum number of complete multipartite subgraphs, with r or fewer parts, needed to partition the edge set, f(G). ... In determining m,(G), w e may assume that no two vertices of G have the same neighbor set. For such reduced graphs G, w e prove that m,(G) 2 log,(n + rl)/r. ... COVERS BY COMPLETE r-PARTITE SUBGRAPHS A graph G is determined by the n X k incidence matrix D of any of its covers (or decompositions) by complete r-partite subgraphs: two vertices of G are adjacent if ...doi:10.1002/(sici)1097-0118(199604)21:4<393::aid-jgt4>3.0.co;2-k fatcat:75h3ajd3jzez7jz5j5mno2jgpi
We classify the results into the following categories: vertex-partitions by monochromatic subgraphs, such as cycles, paths, trees; vertex partition by some kinds of heterochromatic subgraphs; the computational ... complexity of these partition problems; some kinds of large monochromatic and heterochromatic subgraphs. ... Conjecture 1  The vertices of every r-edge colored complete graph K n can be covered by at most r vertex-disjoint monochromatic paths. ...doi:10.1007/s00373-008-0789-5 fatcat:a4j64ccspze2td7cv3f247kf7u
In this paper, we show that the biclique partition number (bp) of a co-chordal (complementary graph of chordal) graph G = (V, E) is less than the number of maximal cliques (mc) of its complementary graph ... Then, an O[|V|(|V|+|E^c|)]-time heuristic is proposed by applying lexicographic breadth-first search. Either heuristic gives us a biclique partition of G with a size of mc(G^c)-1. ... Introduction The biclique partition number (bp) of a graph G is referred to as the least number of complete bipartite (biclique) subgraphs that are required to cover the edges of the graph exactly once ...arXiv:2203.02837v1 fatcat:j6sn6res3jcdjordlune6rnjm4
We prove that for all r≥2 and c>0, every graph of order n with at least cn^r cliques of order r contains a complete r-partite graph with each part of size c^r n . ... This result implies a concise form of the Erdős-Stone theorem. ... Let H * be the subgraph of H induced by the union of the members of S; clearly, H * = K r−1 (s, . . . , s). ...doi:10.1112/blms/bdm093 fatcat:5hjxqwywxfaxhj4cwltsksluwe
We show that any colouring with three colours of the edges of the complete bipartite graph K n,n contains 18 vertex-disjoint monochromatic cycles which together cover all vertices. ... The minimum number of cycles needed for such a covering is five, and we show that this lower bound is asymptotically true. This extends known results for complete graphs. ... Introduction Cycle partitioning complete graphs Given an arbitrary colouring of the edges of a graph G with r colours, we are interested in determining the smallest number of monochromatic cycles that ...doi:10.1016/j.endm.2015.06.106 fatcat:4qf6yjrpxfastnoljsefmifgcm
The r-th iterated line graph L^r(G) of a graph G is defined by: (i) L^0(G) = G and (ii) L^r(G) = L(L^(r- 1)(G)) for r > 0, where L(G) denotes the line graph of G. ... The NP-hard Eulerian Steiner Subgraph problem takes as input a graph G and a specified subset K of terminal vertices of G and asks if G has an Eulerian (that is: connected, and with all vertices of even ... Moreover, from Lemma 11 we get that P1 t P2 is the partition of X defined by the subgraph Gt0 . 3. v? ∈ V(Gt0 ) holds, and V(Gt0 ) is a vertex cover of graph Gt . 4. ...arXiv:1912.01990v1 fatcat:z2ggfot3sbaphkrmvovbavtu5i
The authors solve a problem suggested by M. Tarsi by determining all minimum length cocycle covers of complete graphs. ... The author finds all ordered 4-tuples (a,b,r,p) for which there exists a decomposition of a complete r-partite graph of order p into two factors with diameters a and b. ...
We prove that any finite simple graph can be covered by three of its odd subgraphs, and we construct an infinite sequence of graphs where an edge-disjoint covering by three odd subgraphs is not possible ... I also thank László Pyber for his advices and guidance in the domain of covering problems. ... CCC 2 JOURNAL OF GRAPH THEORY Theorem 2. Every finite simple graph can be covered by three odd subgraphs. Example. ...doi:10.1002/jgt.20170 fatcat:doh4bxbg75aj5af6wg7aksaka4
In a landmark paper, Erdős et al. (1991)  proved that if G is a complete graph whose edges are colored with r colors then the vertex set of G can be partitioned into at most cr 2 log r monochromatic ... Sárközy extended this result to non-complete graphs, and Sárközy and Selkow extended it to k-regular subgraphs. ... The first author's research was supported in part by OTKA Grant No. K68322, by a János Bolyai Research Scholarship and by NSF Grant DMS-0968699. ...doi:10.1016/j.disc.2011.05.031 fatcat:qt4oh3cfkzgujlavsqtj3uvxj4
A graph G is divisible by ¢ if its edge set can be partitioned into t subsets such that the subgraphs (called factors) induced by the subsets are all isomorphic. ... A G-decomposition of the complete graph K,, is a partition of its edge set into subsets such that each subset induces a graph isomorphic to G. ...
r colors then the vertex set of K n can be partitioned into at most 100r log r + 2rk vertex disjoint connected monochromatic k-regular subgraphs and vertices. ... Improving a result of Sárközy and Selkow, we show that for all integers r, k ≥ 2 there exists a constant n 0 = n 0 (r, k ) such that if n ≥ n 0 and the edges of the complete graph K n are colored with ... In every r-coloring of the edges of a complete graph, its vertex set can be partitioned into r monochromatic cycles. ...doi:10.1002/jgt.21661 fatcat:x2xnnrznw5dcnkgwsfnecr44bm
The concept of coloring a graph has been shown to be subsumed by that of an homomorphism. ... This led in  to the definition of a complete n-coloring of a graph G and suggested therefore a new invariant, which we now call the "achromatic number" 4,(G). ... V,,} of the set of points V of a graph G is a K-partition of order m if every induced subgraph (Vi> is complete. ...doi:10.1016/s0021-9800(70)80072-2 fatcat:tjph5jweyfgh7jza7qa5lxfb5u
We formulate the following conjecture: Let n ≥ 5 be an arbitrary positive odd integer; then, in any two-coloring of the edges of the complete 5-partite graph K((n−1)/2, (n−1)/2, (n−1)/2, (n−1)/2, 1) there ... is a monochromatic C n , a cycle of length n. ... One can easily complete the matching M R ∪ M B to a matching M covering almost all vertices of V (H ) (here we can apply Lemma 4 again for the 2-colored partite graph spanned by V (H )\(V (M R )∪ V (M ...doi:10.1002/jgt.20364 fatcat:xqtavuut7jdqzijpqafaoavxne
A collection of n subgraphs of the complete graph K, is an orthogonal double cover if every edge of the complete graph lies in exactly two of the subgraphs, and any two subgraphs have exactly one common ... A &-partition of a graph H is a partition of V(H) into induced subgraphs isomorphic to members of the family ¥. ...
The problem of determining whether a graph G contains a threshold subgraph containing at least h edges is shown to be NP-complete if h is part of the input as the problems of minimum threshold completion ... , weighted 2-threshold partition and weighted 2-threshold covering. ... This implies directly that the problem of the minimum completion of a graph to obtain a threshold graph is NP-complete and we also show that the problems of finding a partition or a covering of weight ...doi:10.1016/0166-218x(94)90214-3 fatcat:62mrfm5cdfdepm2qtahxk4wp4y
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