Cycles, Disjoint Spanning Trees and Orientations of Graphs

A graph G is hamiltonian-connected if any two of its vertices are connected by a Hamilton path (a path including every vertex of G); and G is s-hamiltonian-connected if the deletion of any vertex subset with at most s vertices results in a hamiltonian-connected graph. We prove that the line graph of a (t + 4)-edge-connected graph is (t + 2)-hamiltonian-connected if and only if it is (t + 5)-connected, and for s ≥ 2 every (s + 5)-connected line graph is s-hamiltonian-connected.;For integers l and k with l \u3e 0, and k ≥ 0, Ch( l, k) denotes the collection of h-edge-connected simple graphs G on n vertices such that for every edge-cut X with 2 ≤ |X| ≤ 3, each component of G -- X has at least (n -- k)/l vertices. We prove that for any integer k \u3e 0, there exists an integer N = N( k) such that for any n ≥ N, any graph G ∈ C2(6, k) on n vertices is supereulerian if and only if G cannot be contracted to a member in a well characterized family of graphs.;An orientation of an undirected graph G is a mod (2 p + 1)-orientation if under this orientation, the net out-degree at every vertex is congruence to zero mod 2p + 1. A graph H is mod (2p + 1)-contractible if for any graph G that contains H as a subgraph, the contraction G/H has a mod (2p + 1)-orientation if and only if G has a mod (2p + 1)-orientation (thus every mod (2p + 1)-contractible graph has a mod (2p + 1)-orientation). Jaeger in 1984 conjectured that every (4p)-edge-connected graph has a mod (2p + 1)-orientation. It has also been conjectured that every (4p + 1)-edge-connected graph is mod (2 p + 1)-contractible. We investigate graphs that are mod (2 p + 1)-contractible, and as applications, we prove that a complete graph Km is (2p + 1)-contractible if and only if m ≥ 4p + 1; that every (4p -- 1)-edge-connected K4-minor free graph is mod (2p + 1)-contractible, which is best possible in the sense that there are infinitely many (4p -- 2)-edge-connected K4-minor free graphs that are not mod (2p + 1)-contractible; and that every (4p)-connected chordal graph is mod (2p + 1)-contractible. We also prove that the above conjectures on line graphs would imply the truth of the conjectures in general, and that if G has a mod (2p + 1)-orientation and delta(G) ≥ 4p, then L(G) also has a mod (2p + 1)-orientation.;The design of an n processor network with given number of connections from each processor and with a desirable strength of the network can be modelled as a degree sequence realization problem with certain desirable graphical properties. A nonincreasing sequence d = ( d1, d2, ···, dn) is graphic if there is a simple graph G with degree sequence d. It is proved that for a positive integer k, a graphic nonincreasing sequence d has a simple realization G which has k-edge-disjoint spanning trees if and only if either both n = 1 and d1 = 0, or n ≥ 2 and both dn ≥ k and i=1n di ≥ 2k(n -- 1).;We investigate the emergence of specialized groups in a swarm of robots, using a simplified version of the stick-pulling problem [56], where the basic task requires the collaboration of two robots in asymmetric roles. We expand our analytical model [57] and identify conditions for optimal performance for a swarm with any number of species. We then implement a distributed adaptation algorithm based on autonomous performance evaluation and parameter adjustment of individual agents. While this algorithm reliably reaches optimal performance, it leads to unbounded parameter distributions. Results are improved by the introduction of a direct parameter exchange mechanism between selected high- and low-performing agents. The emerging parameter distributions are bounded and fluctuate between tight unimodal and bimodal profiles. Both the unbounded optimal and the bounded bimodal distributions represent partitions of the swarm into two specialized groups

Spanning trees with few branch vertices

A branch vertex in a tree is a vertex of degree at least three. We prove that, for all $s\geq 1$, every connected graph on $n$ vertices with minimum degree at least $(\frac{1}{s+3}+o(1))n$ contains a spanning tree having at most $s$ branch vertices. Asymptotically, this is best possible and solves, in less general form, a problem of Flandrin, Kaiser, Ku\u{z}el, Li and Ryj\'a\u{c}ek, which was originally motivated by an optimization problem in the design of optical networks.Comment: 20 pages, 2 figures, to appear in SIAM J. of Discrete Mat

Embedding large subgraphs into dense graphs

What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F-packing. The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy and Szemeredi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F-packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved

Almost all trees are almost graceful

The Graceful Tree Conjecture of Rosa from 1967 asserts that the vertices of each tree T of order n can be injectively labeled by using the numbers {1,2,…,n} in such a way that the absolute differences induced on the edges are pairwise distinct. We prove the following relaxation of the conjecture for each γ>0 and for all n>n 0(γ). Suppose that (i) the maximum degree of T is bounded by (Formula presented.)), and (ii) the vertex labels are chosen from the set {1,2,…,⌈(1+γ)n⌉}. Then there is an injective labeling of V(T) such that the absolute differences on the edges are pairwise distinct. In particular, asymptotically almost all trees on n vertices admit such a labeling. The proof proceeds by showing that a certain very natural randomized algorithm produces a desired labeling with high probability

On neighbour sum-distinguishing {0,1}\{0,1\}-edge-weightings of bipartite graphs

Let $S$ be a set of integers. A graph G is said to have the S-property if there exists an S-edge-weighting $w : E(G) \rightarrow S$ such that any two adjacent vertices have different sums of incident edge-weights. In this paper we characterise all bridgeless bipartite graphs and all trees without the $\{0,1\}$-property. In particular this problem belongs to P for these graphs while it is NP-complete for all graphs.Comment: Journal versio

Linearly many rainbow trees in properly edge-coloured complete graphs

A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. The study of rainbow decompositions has a long history, going back to the work of Euler on Latin squares. In this paper we discuss three problems about decomposing complete graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantine’s Conjecture, and the Kaneko-Kano-Suzuki Conjecture. We show that in every proper edge-colouring of Kn there are 10−6n edge-disjoint spanning isomorphic rainbow trees. This simultaneously improves the best known bounds on all these conjectures. Using our method we also show that every properly (n − 1)-edge-coloured Kn has n/9 − 6 edge-disjoint rainbow trees, giving further improvement on the Brualdi-Hollingsworth Conjectur