Sequence variations of the 1-2-3 Conjecture and irregularity strength

Karonski, Luczak, and Thomason (2004) conjectured that, for any connected graph G on at least three vertices, there exists an edge weighting from {1,2,3} such that adjacent vertices receive different sums of incident edge weights. Bartnicki, Grytczuk, and Niwcyk (2009) made a stronger conjecture, that each edge's weight may be chosen from an arbitrary list of size 3 rather than {1,2,3}. We examine a variation of these conjectures, where each vertex is coloured with a sequence of edge weights. Such a colouring relies on an ordering of the graph's edges, and so two variations arise -- one where we may choose any ordering of the edges and one where the ordering is fixed. In the former case, we bound the list size required for any graph. In the latter, we obtain a bound on list sizes for graphs with sufficiently large minimum degree. We also extend our methods to a list variation of irregularity strength, where each vertex receives a distinct sequence of edge weights.Comment: Accepted to Discrete Mathematics and Theoretical Computer Scienc

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

Further Evidence Towards the Multiplicative 1-2-3 Conjecture

The product version of the 1-2-3 Conjecture, introduced by Skowronek-Kazi{\'o}w in 2012, states that, a few obvious exceptions apart, all graphs can be 3-edge-labelled so that no two adjacent vertices get incident to the same product of labels. To date, this conjecture was mainly verified for complete graphs and 3-colourable graphs. As a strong support to the conjecture, it was also proved that all graphs admit such 4-labellings. In this work, we investigate how a recent proof of the multiset version of the 1-2-3 Conjecture by Vu{\v c}kovi{\'c} can be adapted to prove results on the product version. We prove that 4-chromatic graphs verify the product version of the 1-2-3 Conjecture. We also prove that for all graphs we can design 3-labellings that almost have the desired property. This leads to a new problem, that we solve for some graph classes

On asymptotic confirmation of the Faudree-Lehel Conjecture on the irregularity strength of graphs

We call a multigraph irregular if it has pairwise distinct vertex degrees. No nontrivial (simple) graph is thus irregular. The irregularity strength of a graph $G$, $s(G)$, is a specific measure of the ``level of irregularity‘‘ of $G$. It might be defined as the least $k$ such that one may obtain an irregular multigraph of $G$ by multiplying any selected edges of $G$, each into at most $k$ its copies. In other words, $s(G)$ is the least $k$ admitting a $\{1,2,\ldots,k\}$-weighting of the edges of $G$ assuring distinct weighted degrees for all the vertices, where the weighted degree of a vertex is the sum of its incident weights. The most well-known open problem concerning this graph invariant is the conjecture posed in 1987 by Faudree and Lehel that there exists an absolute constant $C$ such that $s(G)\leq \frac{n}{d}+C$ for each $d$-regular graph $G$ with $n$ vertices and $d\geq 2$, whereas a straightforward counting argument implies that $s(G)\geq \frac{n}{d}+\frac{d-1}{d}$. Until very recently this conjecture had remained widely open. We shall discuss recent results confirming it asymptotically, up to a lower order term. If time permits we shall also mention a few related problems, such as the 1-2-3 Conjecture or the concept of irregular subgraphs, introduced recently by Alon and Wei, and progress in research concerning these