Diagonal forms of incidence matrices associated with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>t</mml:mi></mml:math>-uniform hypergraphs

Wilson, R.; Wong, Tony W. H.

doi:10.1016/j.ejc.2013.06.032

Cited by 8 publications

(16 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This quest for a monochromatic spanning tree has been modified in various ways which turn simple results like this into interesting research questions. For example, one can let the colours be 0 and 1 and then require that the sum of the colours on the edges of the spanning tree is even, making it a problem in zero-sum Ramsey-theory over Z 2 , which is completely solved [6,17,16,43]. The problem now takes on a more decidedly Ramsey-theoretic nature because the right question to ask would be: is there an N such that, for all n > N , n odd, any 0-1 colouring of E(K n ) contains a spanning tree such that the sum of the colours on the edges of the spanning tree is even.…”

Section: Introductionmentioning

confidence: 99%

On Zero-Sum Spanning Trees and Zero-Sum Connectivity

Caro¹,

Hansberg²,

Lauri³

et al. 2022

Electron. J. Combin.

View full text Add to dashboard Cite

We consider $2$-colourings $f : E(G) \rightarrow \{ -1 ,1 \}$ of the edges of a graph $G$ with colours $-1$ and $1$ in $\mathbb{Z}$. A subgraph $H$ of $G$ is said to be a zero-sum subgraph of $G$ under $f$ if $f(H) := \sum_{e\in E(H)} f(e) =0$. We study the following type of questions, in several cases obtaining best possible results: Under which conditions on $|f(G)|$ can we guarantee the existence of a zero-sum spanning tree of $G$? The types of $G$ we consider are complete graphs, $K_3$-free graphs, $d$-trees, and maximal planar graphs. We also answer the question of when any such colouring contains a zero-sum spanning path or a zero-sum spanning tree of diameter at most $3$, showing in passing that the diameter-$3$ condition is best possible. Finally, we give, for $G = K_n$, a sharp bound on $|f(K_n)|$ by which an interesting zero-sum connectivity property is forced, namely that any two vertices are joined by a zero-sum path of length at most $4$. One feature of this paper is the proof of an Interpolation Lemma leading to a Master Theorem from which many of the above results follow and which can be of independent interest.

show abstract

Section: Introductionmentioning

confidence: 99%

On Zero-Sum Spanning Trees and Zero-Sum Connectivity

Caro¹,

Hansberg²,

Lauri³

et al. 2022

Electron. J. Combin.

View full text Add to dashboard Cite

show abstract

“…The following special case of our work to follow settles a question of Cameron and Cioabǎ in [1]. Independently, this can be derived from Wilson and Wong's Smith forms [13] of certain related inclusion matrices, although our proof is somewhat more direct.…”

Section: Introductionmentioning

confidence: 75%

“…to distinguish inequivalent Hadamard matrices or to spectral properties of strongly regular graphs. Wilson and Wong [13] use the Smith form to study the module generated by copies of G for simple graphs and certain special multigraphs G.…”

Section: 2mentioning

confidence: 99%

“…Theorem 1.1 (see also [13]). Let G be a simple graph with n vertices, e > 0 edges, and vertex degrees d 1 , .…”

Section: Introductionmentioning

confidence: 96%

“…In the special case when A is a (nonzero) {0, 1}-valued symmetric matrix with zeros on the diagonal, it is the adjacency matrix of a simple graph, say G. Our problem then reduces to finding the least positive integer λ such that the λ-fold complete graph λK n admits a 'signed' edge-decomposition into copies of G. This question has been studied before [1,3,12,13], although to our knowledge an explicit formula for λ has not been written down in all cases. As observed in these references, especially [1,12], this λ is also the eventual period for multiples of the complete graph to admit an edge-decomposition (in the traditional 'unsigned' setting) into copies of G.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Balancing permuted copies of multigraphs and integer matrices

Valle¹,

Dukes²

2022

Preprint

View full text Add to dashboard Cite

Given a square matrix A over the integers, we consider the Z-module M A generated by the set of all matrices that are permutation-similar to A. Motivated by analogous problems on signed graph decompositions and block designs, we are interested in the completely symmetric matrices aI + bJ belonging to M A . We give a relatively fast method to compute a generator for such matrices, avoiding the need for a very large canonical form over Z. We consider several special cases in detail. In particular, the problem for symmetric matrices answers a question of Cameron and Cioabǎ on determining the eventual period for integers λ such that the λ-fold complete graph λKn has an edge-decomposition into a given (multi)graph.

show abstract