Monochromatic Components Research Articles

Large monochromatic components in expansive hypergraphs

AbstractA result of Gyárfás [12] exactly determines the size of a largest monochromatic component in an arbitrary $r$ -colouring of the complete $k$ -uniform hypergraph $K_n^k$ when $k\geq 2$ and $k\in \{r-1,r\}$ . We prove a result which says that if one replaces $K_n^k$ in Gyárfás’ theorem by any ‘expansive’ $k$ -uniform hypergraph on $n$ vertices (that is, a $k$ -uniform hypergraph $G$ on $n$ vertices in which $e(V_1, \ldots, V_k)\gt 0$ for all disjoint sets $V_1, \ldots, V_k\subseteq V(G)$ with $|V_i|\gt \alpha$ for all $i\in [k]$ ), then one gets a largest monochromatic component of essentially the same size (within a small error term depending on $r$ and $\alpha$ ). As corollaries we recover a number of known results about large monochromatic components in random hypergraphs and random Steiner triple systems, often with drastically improved bounds on the error terms.Gyárfás’ result is equivalent to the dual problem of determining the smallest possible maximum degree of an arbitrary $r$ -partite $r$ -uniform hypergraph $H$ with $n$ edges in which every set of $k$ edges has a common intersection. In this language, our result says that if one replaces the condition that every set of $k$ edges has a common intersection with the condition that for every collection of $k$ disjoint sets $E_1, \ldots, E_k\subseteq E(H)$ with $|E_i|\gt \alpha$ , there exists $(e_1, \ldots, e_k)\in E_1\times \cdots \times E_k$ such that $e_1\cap \cdots \cap e_k\neq \emptyset$ , then the smallest possible maximum degree of $H$ is essentially the same (within a small error term depending on $r$ and $\alpha$ ). We prove our results in this dual setting.

Monochromatic components with many edges

Monochromatic components with many edges

Asymptotic dispersion engineering for ultra-broadband meta-optics

Dispersion decomposes compound light into its monochromatic components, which is detrimental to broadband imaging but advantageous for spectroscopic applications. Metasurfaces provide a unique path to modulate the dispersion by adjusting structural parameters on a two-dimensional plane. However, conventional linear phase compensation does not adequately match the meta-unit’s dispersion characteristics with required complex dispersion, hindering at-will dispersion engineering over a very wide bandwidth particularly. Here, we propose an asymptotic phase compensation strategy for ultra-broadband dispersion-controlled metalenses. Metasurfaces with extraordinarily high aspect ratio nanostructures have been fabricated for arbitrary dispersion control in ultra-broad bandwidth, and we experimentally demonstrate the single-layer achromatic metalenses in the visible to infrared spectrum (400 nm~1000 nm, NA = 0.164). Our proposed scheme provides a comprehensive theoretical framework for single-layer meta-optics, allowing for arbitrary dispersion manipulation without bandwidth restrictions. This development is expected to have significant applications in ultra-broadband imaging and chromatography detection, among others.

Large monochromatic components in hypergraphs with large minimum codegree

AbstractA result of Gyárfás says that for every 3‐coloring of the edges of the complete graph , there is a monochromatic component of order at least , and this is best possible when 4 divides . Furthermore, for all and every ‐coloring of the edges of the complete ‐uniform hypergraph , there is a monochromatic component of order at least and this is best possible for all . Recently, Guggiari and Scott and independently Rahimi proved a strengthening of the graph case in the result above which says that the same conclusion holds if is replaced by any graph on vertices with minimum degree at least ; furthermore, this bound on the minimum degree is best possible. We prove a strengthening of the case in the result above which says that the same conclusion holds if is replaced by any ‐uniform hypergraph on vertices with minimum ‐degree at least ; furthermore, this bound on the ‐degree is best possible.

Color Reproduction by Multi-Wavelength Bragg Diffraction of White Light.

Accurate color reproduction is highly important in multiple industrial, biomedical and scientific applications. Versatile and tunable light sources with high color-rendering quality are very much in demand. In this study, we demonstrate the feasibility of multi-wavelength Bragg diffraction of light for this task. Tuning the frequencies and amplitudes of bulk acoustic waves in the birefringent crystal demonstrates high precision in setting the number, wavelengths and intensities of the monochromatic components necessary to reproduce a specific color assigned according to its coordinates in the CIE XYZ 1931 space. We assembled a setup based on multi-bandpass acousto-optic (AO) filtration of white light and verified the reproduced color balance in multiple experiments. The proposed approach delivers almost full coverage of the CIE XYZ 1931 space and facilitates building compact color reproduction systems (CRSs) for various purposes.

Clustered coloring of graphs with bounded layered treewidth and bounded degree

The clustering of a graph coloring is the maximum size of monochromatic components. This paper studies colorings with bounded clustering in graph classes with bounded layeredtreewidth, which include planar graphs, graphs of bounded Euler genus, graphs embeddable on a fixed surface with a bounded number of crossings per edge, map graphs, amongst other examples. Our main theorem says that every graph with layered treewidth at most k and with maximum degree at most Δ is 3-colorable with clustering O(k19Δ37). This is the first known polynomial bound on the clustering. This greatly improves upon a corresponding result of Esperet and Joret for graphs of bounded genus.

Suppression of stimulated Raman scattering in plasma by an ultra-wideband stochastic phase low-coherence laser

Suppression of the stimulated Raman scattering (SRS) by a stochastic phase low-coherence laser (SPLL) in homogeneous plasma is investigated by theoretical analysis and one-dimensional particle-in-cell (PIC) simulation. A simple model is established, in which the SPLL is modelled as a combination of a monochromatic laser and a broadband laser. When the phase randomness increases, the bandwidth of the SPLL is broadened and the energy proportion of the monochromatic laser component is reduced. PIC simulation shows that the SPLL can effectively suppress SRS and hot electron generation. Various phenomena in the nonlinear process, such as the nonlinear frequency shift and the competition between forward-scattering and back-scattering modes, are explained in detail.

Response spectra for transient pore-pressure in a sandy seabed under random waves: Frequency-filtering effect

The real state of ocean waves is random in nature with various monochromatic components. Transient pore pressures in a sandy seabed under the action of the random-waves with JONSWAP spectrum and the regular-waves were physically modeled, respectively. The surface elevations and the corresponding transient pore pressures in the fine-sand were measured simultaneously in a wave flume. The frequency-filtering phenomenon was observed by Fourier analysis for random-wave induced pore pressure responses. In the Fourier energy spectra, both a primary-peak (high-frequency) and a secondary-peak (low-frequency) were identified within the relatively shallow soil layer. But the primary-peak kept declining rapidly, meanwhile the secondary-peak attenuated much more slowly with increasing soil depth, which can eventually lead to a shift between the high-frequency peak and the low-frequency one. As a consequence, the upgraded primary-peak frequencies of the dual-peaked spectra at the deeper soil layer would become smaller than the original significant frequencies, indicating that the pore pressure induced by long-wave components is of vital significance there. Furthermore, a filtering-depth parameter is introduced to characterize the frequency-filtering effect. Given the pore pressure at a certain soil depth, the quantitative relationship between the primary-peak and the secondary-peak can be predicted for an arbitrary soil depth.

Crumby colorings — Red-blue vertex partition of subcubic graphs regarding a conjecture of Thomassen

Thomassen formulated the following conjecture: Every 3-connected cubic graph has a red-blue vertex coloring such that the blue subgraph has maximum degree at most 1 (that is, it consists of a matching and some isolated vertices) and the red subgraph has minimum degree at least 1 and contains no 3-edge path. Since all monochromatic components are small in this coloring and there is a certain irregularity, we call such a coloring crumby. Recently, Bellitto, Klimošová, Merker, Witkowski and Yuditsky [2] constructed an infinite family refuting the above conjecture. Their prototype counterexample is 2-connected, planar, but contains a K4-minor and also a 5-cycle. This leaves the above conjecture open for some important graph classes: outerplanar graphs, K4-minor-free graphs, bipartite graphs. In this regard, we prove that 2-connected outerplanar graphs, subdivisions of K4 and 1-subdivisions of cubic graphs admit crumby colorings. A subdivision of G is genuine if every edge is subdivided at least once. We show that every genuine subdivision of any subcubic multigraph admits a crumby coloring. We slightly generalize some of these results and formulate a few conjectures.

Covering random graphs with monochromatic trees

AbstractGiven an ‐edge‐colored complete graph , how many monochromatic connected components does one need in order to cover its vertex set? This natural question is a well‐known essentially equivalent formulation of the classical Ryser's conjecture which, despite a lot of attention over the last 50 years, still remains open. A number of recent papers consider a sparse random analogue of this question, asking for the minimum number of monochromatic components needed to cover the vertex set of an ‐edge‐colored random graph . Recently, Bucić, Korándi, and Sudakov established a connection between this problem and a certain Helly‐type local to global question for hypergraphs raised about 30 years ago by Erdős, Hajnal, and Tuza. We identify a modified version of the hypergraph problem which controls the answer to the problem of covering random graphs with monochromatic components more precisely. To showcase the power of our approach, we essentially resolve the 3‐color case by showing that is a threshold at which point three monochromatic components are needed to cover all vertices of a 3‐edge‐colored random graph, answering a question posed by Kohayakawa, Mendonça, Mota, and Schülke. Our approach also allows us to determine the answer in the general ‐edge colored instance of the problem, up to lower order terms, around the point when it first becomes bounded, answering a question of Bucić, Korándi, and Sudakov.

Improved bound for improper colourings of graphs with no odd clique minor

AbstractStrengthening Hadwiger’s conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$ -minor is properly $(t-1)$ -colourable. This is known as the Odd Hadwiger’s conjecture. We prove a relaxation of the above conjecture, namely we show that every graph with no odd $K_t$ -minor admits a vertex $(2t-2)$ -colouring such that all monochromatic components have size at most $\lceil \frac{1}{2}(t-2) \rceil$ . The bound on the number of colours is optimal up to a factor of $2$ , improves previous bounds for the same problem by Kawarabayashi (2008, Combin. Probab. Comput.17 815–821), Kang and Oum (2019, Combin. Probab. Comput.28 740–754), Liu and Wood (2021, arXiv preprint, arXiv:1905.09495), and strengthens a result by van den Heuvel and Wood (2018, J. Lond. Math. Soc.98 129–148), who showed that the above conclusion holds under the more restrictive assumption that the graph is $K_t$ -minor-free. In addition, the bound on the component-size in our result is much smaller than those of previous results, in which the dependency on $t$ was given by a function arising from the graph minor structure theorem of Robertson and Seymour. Our short proof combines the method by van den Heuvel and Wood for $K_t$ -minor-free graphs with some additional ideas, which make the extension to odd $K_t$ -minor-free graphs possible.

Immersion and clustered coloring

Hadwiger and Hajós conjectured that for every positive integer t, Kt+1-minor free graphs and Kt+1-topological minor free graphs are properly t-colorable, respectively. Clustered coloring version of these two conjectures which only require monochromatic components to have bounded size has been extensively studied. In this paper we consider the clustered coloring version of the immersion-variant of Hadwiger's and Hajós' conjecture proposed by Lescure and Meyniel and independently by Abu-Khzam and Langston. We determine the minimum number of required colors for H-immersion free graphs, for any fixed graph H, up to a small additive absolute constant. Our result is tight for infinitely many graphs H.A key machinery developed in this paper is a lemma that reduces a clustering coloring problem on graphs to the one on the torsos of their tree-cut decomposition or tree-decomposition. A byproduct of this machinery is a unified proof of a result of Alon, Ding, Oporowski and Vertigan and a result of the author and Oum about clustered coloring graphs of bounded maximum degree in minor-closed families.

Clustered colouring of graph classes with bounded treedepth or pathwidth

AbstractThe clustered chromatic number of a class of graphs is the minimum integer $k$ such that for some integer $c$ every graph in the class is $k$ -colourable with monochromatic components of size at most $c$ . We determine the clustered chromatic number of any minor-closed class with bounded treedepth, and prove a best possible upper bound on the clustered chromatic number of any minor-closed class with bounded pathwidth. As a consequence, we determine the fractional clustered chromatic number of every minor-closed class.

Ban–Linial's Conjecture and treelike snarks

AbstractA bridgeless cubic graph is said to have a 2‐bisection if there exists a 2‐vertex‐colouring of (not necessarily proper) such that: (i) the colour classes have the same cardinality, and (ii) the monochromatic components are either an isolated vertex or an edge. In 2016, Ban and Linial conjectured that every bridgeless cubic graph, apart from the well‐known Petersen graph, admits a 2‐bisection. In the same paper it was shown that every Class I bridgeless cubic graph admits such a bisection. The Class II bridgeless cubic graphs which are critical to many conjectures in graph theory are known as snarks, in particular, those with excessive index at least 5, that is, whose edge set cannot be covered by four perfect matchings. Moreover, Esperet et al. state that a possible counterexample to Ban–Linial's Conjecture must have circular flow number at least 5. The same authors also state that although empirical evidence shows that several graphs obtained from the Petersen graph admit a 2‐bisection, they can offer nothing in the direction of a general proof. Despite some sporadic computational results, until now, no general result about snarks having excessive index and circular flow number both at least 5 has been proven. In this study we show that treelike snarks, which are an infinite family of snarks heavily depending on the Petersen graph and with both their circular flow number and excessive index at least 5, admit a 2‐bisection.

Spatial distribution and time evolution of metal-containing plasma of a low-current atmospheric pressure discharge with magnesium cathode

The spatial intensity distribution and temporal dynamics of the plasma generated by an atmospheric pressure discharge with magnesium cathode in an argon flow are investigated in a coaxial geometry discharge system. The repetition rate of unipolar pulses was 56 kHz and the pulse duration was 12 μs. The steady-state amplitude of the discharge current was 100 mA at a voltage of about 130 V. Under this operating mode, a local melting of the active cathode surface took place. The evaporated magnesium atoms were captured by the working gas flow and formed a green glow plume around the positive discharge column outside the anode nozzle. The image of the plasma formation was projected onto the entrance slit of the monochromator. The spatial distribution of the radiation intensity and evolution in time of its selected monochromatic components were measured. The radiation spectrum contained groups of ion and magnesium atom lines with wavelengths of 285.21 nm (singlet resonant Mg I); 383.08, 383.36, 383.9 nm (triplet Mg I); 517.3, 517.5, 518.1 nm (triplet Mg I). The results of this work are promising with regard to studying open-type spontaneous radiation sources, as well as the generation of combined gas-metal plasma flows at atmospheric pressure.

Clustered variants of Hajós' conjecture

Hajós conjectured that every graph containing no subdivision of the complete graph Ks+1 is properly s-colorable. This conjecture was disproved by Catlin. Indeed, the maximum chromatic number of such graphs is Ω(s2/log⁡s). We prove that O(s) colors are enough for a weakening of this conjecture that only requires every monochromatic component to have bounded size (so-called clustered coloring).Our approach leads to more results, many of which only require a much weaker assumption that forbids an ‘almost (⩽1)-subdivision’ (where at most one edge is subdivided more than once). This assumption is best possible, since no bound on the number of colors exists unless we allow at least one edge to be subdivided arbitrarily many times. We prove the following (where s⩾2):1.Graphs of bounded treewidth and with no almost (⩽1)-subdivision of Ks+1 are s-choosable with bounded clustering.2.For every graph H, graphs with no H-minor and no almost (⩽1)-subdivision of Ks+1 are (s+1)-colorable with bounded clustering.3.For every graph H of maximum degree at most d, graphs with no H-subdivision and no almost (⩽1)-subdivision of Ks+1 are max⁡{s+3d−5,2}-colorable with bounded clustering.4.For every graph H of maximum degree d, graphs with no Ks,t subgraph and no H-subdivision are max⁡{s+3d−4,2}-colorable with bounded clustering.5.Graphs with no Ks+1-subdivision are (4s−5)-colorable with bounded clustering. The first result is tight and shows that the clustered analogue of Hajós' conjecture is true for graphs of bounded treewidth. The second result implies an upper bound for the clustered version of Hadwiger's conjecture that is only one color away from the known lower bound, and shows that the number of colors is independent of the forbidden minor. The final result is the first O(s) bound on the clustered chromatic number of graphs with no Ks+1-subdivision.

Large monochromatic components of small diameter

AbstractGyárfás conjectured in 2011 that every ‐edge‐colored contains a monochromatic component of bounded (“perhaps three”) diameter on at least vertices. Letzter proved this conjecture with diameter four. In this note we improve the result in the case of : We show that in every 3‐edge‐coloring of either there is a monochromatic component of diameter at most three on at least vertices or every color class is spanning and has diameter at most four.

Covering random graphs by monochromatic trees

Given an $r$-edge-coloured complete graph $K_n$, how many monochromatic connected components does one need in order to cover its vertex set? This natural question is a well-known essentially equivalent formulation of the classical Ryser's conjecture which, despite a lot of attention over the last 50 years, still remains open. A number of recent papers consider a sparse random analogue of this question, asking for the minimum number of monochromatic components needed to cover the vertex set of an $r$-edge-coloured random graph $\mathcal{G}(n,p)$. Recently, Bucic, Korandi and Sudakov established a connection between this problem and a certain Helly-type local to global question for hypergraphs raised about 30 years ago by Erdős, Hajnal and Tuza. We identify a modified version of the hypergraph problem which controls the answer to the problem of covering random graphs with monochromatic components more precisely. To showcase the power of our approach, we essentially resolve the $3$-colour case by showing that $(\log n / n)^{1/4}$ is a threshold at which point three monochromatic components are needed to cover all vertices of a $3$-edge-coloured random graph, answering a question posed by Kohayakawa, Mendonca, Mota and Schulke. Our approach also allows us to determine the answer in the general $r$-edge coloured instance of the problem, up to lower order terms, around the point when it first becomes bounded, answering a question of Bucic, Korandi and Sudakov.

Monochromatic diameter-2 components in edge colorings of the complete graph

Gyarfas conjectured that in every r -edge-coloring of the complete graph K n there is a monochromatic component on at least n ∕ ( r − 1 ) vertices which has diameter at most 3. We show that for r = 3 , 4 , 5 and 6 a diameter of 3 is best possible in this conjecture, constructing colorings where every monochromatic diameter-2 subgraph has strictly less than n ∕ ( r − 1 ) vertices.

Clustered 3-colouring graphs of bounded degree

AbstractA (not necessarily proper) vertex colouring of a graph hasclustering cif every monochromatic component has at mostcvertices. We prove that planar graphs with maximum degree$\Delta$are 3-colourable with clustering$O(\Delta^2)$. The previous best bound was$O(\Delta^{37})$. This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree$\Delta$that exclude a fixed minor are 3-colourable with clustering$O(\Delta^5)$. The best previous bound for this result was exponential in$\Delta$.