Modular Irregular Labeling on Double-Star and Friendship Graphs

A modular irregular graph is a graph that admits a modular irregular labeling. A modular irregular labeling of a graph G of order n is a mapping of the set of edges of the graph to 1,2 , … , k such that the weights of all vertices are different. The vertex weight is the sum of its incident edge labels, and all vertex weights are calculated with the sum modulo n . The modular irregularity strength is the minimum largest edge label such that a modular irregular labeling can be done. In this paper, we construct a modular irregular labeling of two classes of graphs that are biregular; in this case, the regular double-star graph and friendship graph classes are chosen. Since the modular irregularity strength of the friendship graph also holds the minimal irregularity strength, then the labeling is also an irregular labeling with the same strength as the modular case.

A NOTE ON EDGE WEIGHT CHOOSABILITY OF GRAPHS

In 2004, Karoński, Łuczak, and Thomason conjectured that the edges of any connected graph on at least 3 vertices may be weighted from the set {1, 2, 3} so that the vertices are properly colored by the sums of their incident edge weights. Bartnicki, Grytczuk and Niwcyk introduced its list version. Assign to each edge e ∈ E(G) a list of k real numbers, say L(e), and choose a weight w(e) ∈ L(e) for each e ∈ E(G). The resulting function w : E(G) → ⋃e∈E(G) L(e) is called an edge k-list-weighting. Given a graph G, the smallest k such that any assignment of lists of size k to E(G) permits an edge k-list-weighting which is a vertex coloring by sums is denoted by [Formula: see text] and called the edge weight choosability of G. Bartnicki, Grytczuk and Niwcyk conjectured that if G is a nice graph (without a component isomorphic to K2), then [Formula: see text]. There is no known constant K such that [Formula: see text] for any nice graph G. Ben Seamone proved that [Formula: see text] for any nice graph G with maximum degree Δ(G) by using Alon's Combinatorial Nullstellensatz. In this paper, we improve this bound to [Formula: see text].

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

Graph Theory International audience Karonski, Luczak, and Thomason (2004) conjecture 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) make 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 E(G), and so two variations arise - one where we may choose any ordering of E(G) 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.

Mixing Time for a Markov Chain on Cladograms

A cladogram is a tree with labelled leaves and unlabelled degree-3 branchpoints. A certain Markov chain on the set of n-leaf cladograms consists of removing a random leaf (and its incident edge) and re-attaching it to a random edge. We show that the mixing time (time to approach the uniform stationary distribution) for this chain is at least O(n2) and at most O(n3).

Resonant scattering of edge waves by longshore periodic topography

The resonant scattering of topographically trapped, low-mode progressive edge waves by longshore periodic topography is investigated using a multiple-scale expansion of the linear shallow water equations. Coupled evolution equations for the slowly varying amplitudes of incident and scattered edge waves are derived for small-amplitude, periodic depth perturbations superposed on a plane beach. In ‘single-wave scattering’, an incident edge wave is resonantly scattered into a single additional progressive edge wave having the same or different mode number (i.e. longshore wavenumber), and propagating in the same or opposite direction (forward and backward scattering, respectively), as the incident edge wave. Backscattering into the same mode number as the incident edge wave, the analogue of Bragg scattering of surface waves, is a special case. In ‘multi-wave scattering’, simultaneous forward and backward resonant scattering results in several (rather than only one) new progressive edge waves. Analytic solutions are obtained for single-wave scattering and for a special case of multi-wave scattering involving mode-0 and mode-1 edge waves, over perturbed depth regions of both finite and semi-infinite longshore extent. In single-wave backscattering with small (subcritical) detuning (i.e. departure from exact resonance), the incident and backscattered wave amplitudes both decay exponentially with propagation distance over the periodic bathymetry, whereas with large (supercritical) detuning the amplitudes oscillate with distance. In single-wave forward scattering, the wave amplitudes are oscillatory regardless of the magnitude of the detuning. Multi-wave solutions combine aspects of single-wave backward and forward scattering. In both single- and multi-wave scattering, the exponential decay rates and oscillatory wavenumbers of the edge wave amplitudes depend on the detuning. The results suggest that naturally occurring rhythmic features such as beach cusps and crescentic bars are sometimes of large enough amplitude to scatter a significant amount of incident low-mode edge wave energy in a relatively short distance (O(10) topographic wavelengths).