10 research outputs found
On neighbour sum-distinguishing -edge-weightings of bipartite graphs
Let be a set of integers. A graph G is said to have the S-property if
there exists an S-edge-weighting 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
-property. In particular this problem belongs to P for these graphs
while it is NP-complete for all graphs.Comment: Journal versio
Spanning trees without adjacent vertices of degree 2
Albertson, Berman, Hutchinson, and Thomassen showed in 1990 that there exist
highly connected graphs in which every spanning tree contains vertices of
degree 2. Using a result of Alon and Wormald, we show that there exists a
natural number such that every graph of minimum degree at least
contains a spanning tree without adjacent vertices of degree 2. Moreover, we
prove that every graph with minimum degree at least 3 has a spanning tree
without three consecutive vertices of degree 2
On a combination of the 1-2-3 conjecture and the antimagic labelling conjecture
International audienceThis paper is dedicated to studying the following question: Is it always possible to injectively assign the weights 1, ..., |E(G)| to the edges of any given graph G (with no component isomorphic to K2) so that every two adjacent vertices of G get distinguished by their sums of incident weights? One may see this question as a combination of the well-known 1-2-3 Conjecture and the Antimagic Labelling Conjecture. Throughout this paper, we exhibit evidence that this question might be true. Benefiting from the investigations on the Antimagic Labelling Conjecture, we first point out that several classes of graphs, such as regular graphs, indeed admit such assignments. We then show that trees also do, answering a recent conjecture of Arumugam, Premalatha, BaÄa and SemaniÄovĂĄ-FeĆovÄĂkovĂĄ. Towards a general answer to the question above, we then prove that claimed assignments can be constructed for any graph, provided we are allowed to use some number of additional edge weights. For some classes of sparse graphs, namely 2-degenerate graphs and graphs with maximum average degree 3, we show that only a small (constant) number of such additional weights suffices