11 research outputs found
On -chromatic numbers of graphs having bounded sparsity parameters
An -graph is characterised by having types of arcs and types
of edges. A homomorphism of an -graph to an -graph , is a
vertex mapping that preserves adjacency, direction, and type. The
-chromatic number of , denoted by , is the minimum
value of such that there exists a homomorphism of to . The
theory of homomorphisms of -graphs have connections with graph theoretic
concepts like harmonious coloring, nowhere-zero flows; with other mathematical
topics like binary predicate logic, Coxeter groups; and has application to the
Query Evaluation Problem (QEP) in graph database.
In this article, we show that the arboricity of is bounded by a function
of but not the other way around. Additionally, we show that the
acyclic chromatic number of is bounded by a function of , a
result already known in the reverse direction. Furthermore, we prove that the
-chromatic number for the family of graphs with a maximum average degree
less than , including the subfamily of planar graphs
with girth at least , equals . This improves upon previous
findings, which proved the -chromatic number for planar graphs with
girth at least is .
It is established that the -chromatic number for the family
of partial -trees is both bounded below and above by
quadratic functions of , with the lower bound being tight when
. We prove and which improves both known lower bounds and
the former upper bound. Moreover, for the latter upper bound, to the best of
our knowledge we provide the first theoretical proof.Comment: 18 page
Circular choosability
International audienceWe study circular choosability, a notion recently introduced by Mohar and by Zhu. First, we provide a negative answer to a question of Zhu about circular cliques. We next prove that cch(G) = O(ch(G) + ln |V(G)|) for every graph G. We investigate a generalisation of circular choosability, the circular f-choosability, where f is a function of the degrees. We also consider the circular choice number of planar graphs. Mohar asked for the value of Ï„ := sup {cch(G) : G is planar}, and we prove that 68, thereby providing a negative answer to another question of Mohar. We also study the circular choice number of planar and outerplanar graphs with prescribed girth, and graphs with bounded density
Planar Graphs with Homomorphisms to the 9-cycle
We study the problem of finding homomorphisms into odd cycles from planar
graphs with high odd-girth. The Jaeger-Zhang conjecture states that every
planar graph of odd-girth at least admits a homomorphism to the odd
cycle . The case is the well-known Gr\"otzsch's -coloring
theorem. For general , in 2013 Lov\'asz, Thomassen, Wu, and Zhang showed
that it suffices to have odd-girth at least . Improvements are known for
and in [Combinatorica 2017, SIDMA 2020, Combinatorica 2022]. For
we improve this hypothesis by showing that it suffices to have odd-girth
23. Our main tool is a variation on the potential method applied to modular
orientations. This allows more flexibility when seeking reducible
configurations. The same techniques also prove some results on circular
coloring of signed planar graphs.Comment: 24 pages, 4 figure
Modeling and Tuning of Energy Harvesting Device Using Piezoelectric Cantilever Array
Piezoelectric devices have been increasingly investigated as a means of converting ambient vibrations into electrical energy that can be stored and used to power other devices, such as the sensors/actuators, micro-electro-mechanical systems (MEMS) devices, and microprocessor units etc. The objective of this work was to design, fabricate, and test a piezoelectric device to harvest as much power as possible from vibration sources and effectively store the power in a battery.;The main factors determining the amount of collectable power of a single piezoelectric cantilever are its resonant frequency, operation mode and resistive load in the charging circuit. A proof mass was used to adjust the resonant frequency and operation mode of a piezoelectric cantilever by moving the mass along the cantilever. Due to the tiny amount of collected power, a capacitor was suggested in the charging circuit as an intermediate station. To harvest sufficient energy, a piezoelectric cantilever array, which integrates multiple cantilevers in parallel connection, was investigated.;In the past, most prior research has focused on the theoretical analysis of power generation instead of storing generated power in a physical device. In this research, a commercial solid-state battery was used to store the power collected by the proposed piezoelectric cantilever array. The time required to charge the battery up to 80% capacity using a constant power supply was 970 s. It took about 2400 s for the piezoelectric array to complete the same task. Other than harvesting energy from sinusoidal waveforms, a vibration source that emulates a real environment was also studied. In this research the response of a bridge-vehicle system was used as the vibration sources such a scenario is much closer to a real environment compared with typical lab setups
Density and Structure of Homomorphism-Critical Graphs
Let be a graph. A graph is -critical if every proper subgraph of admits a homomorphism to , but itself does not. In 1981, Jaeger made the following conjecture concerning odd-cycle critical graphs: every planar graph of girth at least admits a homomorphism to (or equivalently, has a -circular colouring). The best known result for the case states that every planar graph of girth at least 18 has a homomorphism to . We improve upon this result, showing that every planar graph of girth at least 16 admits a homomorphism to . This is obtained from a more general result regarding the density of -critical graphs. Our main result is that if is a -critical graph with , then . Additionally, we prove several structural lemmas concerning graphs that are -critical, when is a vertex-transitive non-bipartite graph