Academic literature on the topic 'Rainbow perfect matching'

A perfect matchingMin an edge-coloured complete bipartite graphKn,nis rainbow if no pair of edges inMhave the same colour. We obtain asymptotic enumeration results for the number of rainbow perfect matchings in terms of the maximum number of occurrences of each colour. We also consider two natural models of random edge-colourings ofKn,nand show that if the number of colours is at leastn, then there is with high probability a rainbow perfect matching. This in particular shows that almost every square matrix of ordernin which every entry appearsntimes has a Latin transversal.

The anti-Ramsey number ARG,H is the maximum number of colors in an edge-coloring of G such that G contains no rainbow subgraphs isomorphic to H. In this paper, we discuss the anti-Ramsey numbers ARKp1,p2,…,pk,Tn, ARKp1,p2,…,pk,ℳ, and ARKp1,p2,…,pk,C of Kp1,p2,…,pk, where Tn,ℳ, and C denote the family of all spanning trees, the family of all perfect matchings, and the family of all Hamilton cycles in Kp1,p2,…,pk, respectively.

Aharoni and Berger conjectured that every collection of $n$ matchings of size $n+1$ in a bipartite graph contains a rainbow matching of size $n$. This conjecture is related to several old conjectures of Ryser, Brualdi, and Stein about transversals in Latin squares. There have been many recent partial results about the Aharoni-Berger Conjecture. The conjecture is known to hold when the matchings are much larger than $n+1$. The best bound is currently due to Aharoni, Kotlar, and Ziv who proved the conjecture when the matchings are of size at least $3n/2+1$. When the matchings are all edge-disjoint and perfect, the best result follows from a theorem of Häggkvist and Johansson which implies the conjecture when the matchings have size at least $n+o(n)$. In this paper we show that the conjecture is true when the matchings have size $n+o(n)$ and are all edge-disjoint (but not necessarily perfect). We also give an alternative argument to prove the conjecture when the matchings have size at least $\phi n+o(n)$ where $\phi\approx 1.618$ is the Golden Ratio.Our proofs involve studying connectedness in coloured, directed graphs. The notion of connectedness that we introduce is new, and perhaps of independent interest.

Abstract We study the following rainbow version of subgraph containment problems in a family of (hyper)graphs, which generalizes the classical subgraph containment problems in a single host graph. For a collection $\mathit {\mathbf {G}}=\{G_1, G_2,\ldots , G_{m}\}$ of not necessarily distinct k-graphs on the same vertex set $[n]$ , a (sub)graph H on $[n]$ is rainbow if there exists an injection $\varphi : E(H)\rightarrow [m]$ , such that $e\in E(G_{\varphi (e)})$ for each $e\in E(H)$ . Note that if $|E(H)|=m$ , then $\varphi $ is a bijection, and thus H contains exactly one edge from each $G_i$ . Our main results focus on rainbow clique-factors in (hyper)graph systems with minimum d-degree conditions. Specifically, we establish the following: (1) A rainbow analogue of an asymptotical version of the Hajnal–Szemerédi theorem, namely, if $t\mid n$ and $\delta (G_i)\geq (1-\frac {1}{t}+\varepsilon )n$ for each $i\in [\frac {n}{t}\binom {t}{2}]$ , then $\mathit {\mathbf {G}}$ contains a rainbow $K_t$ -factor; (2) Essentially, a minimum d-degree condition forcing a perfect matching in a k-graph also forces rainbow perfect matchings in k-graph systems for $d\in [k-1]$ . The degree assumptions in both results are asymptotically best possible (although the minimum d-degree condition forcing a perfect matching in a k-graph is in general unknown). For (1), we also discuss two directed versions and a multipartite version. Finally, to establish these results, we in fact provide a general framework to attack this type of problem, which reduces it to subproblems with finitely many colors.

Introduction The sensational appearance of kawaii fashion in Tokyo’s Harajuku neighborhood—full of freedom, fun, and frills— has captivated hearts and imaginations worldwide. A key motivational concept for this group is “kawaii” which is commonly translated as “cute” and can also be used to describe things that are “beautiful”, “funny”, “pretty”, “wonderful”, “great”, “interesting”, and “kind” (Yamane 228; Yomota 73; Dale 320). Representations in media such as the styling of Harajuku street model and J-pop star Kyary Pamyu Pamyu, directed by Sebastian Masuda, have helped bring this fashion to a wider audience. Of this vibrant community, decora fashion is perhaps best known with its image well documented in in street-fashion magazines such as Shoichi Aoki’s FRUiTS (1997–2017), Websites such as Tokyo Fashion (2000–present), and in magazines like KERA (1998–2017). In particular, decora fashion captures the “do-it-yourself” approach for which Harajuku is best known for (Yagi 17). In this essay we draw on New Materialism to explore the ways in which decora fashion practitioners form kawaii affective assemblages with the objects they collect and transform into fashion items. We were motivated to pursue this research to build on other qualitative studies that aimed to include the voices of practitioners in accounts of their lifestyles (e.g. Nguyen; Monden; Younker) and respond to claims that kawaii fashion is a form of infantile regression. We—an Australian sociologist and kawaii fashion practitioner, a Japanese decora fashion practitioner and Harajuku street model, and a Japanese former owner of a tearoom in Harajuku—have used an action-led participatory research method to pool our expertise. In this essay we draw on both a New Materialist analysis of our own fashion practices, a 10-year longitudinal study of Harajuku (2012–2022), as well as interviews with twelve decora fashion practitioners in 2020. What Is Decora Fashion? Decora is an abbreviation of “decoration”, which reflects the key aesthetic commitment of the group to adorn their bodies with layers of objects, accessories, and stickers. Decora fashion uses bright clothing from thrift stores, layers of handmade and store-bought accessories, and chunky platform shoes or sneakers. Practitioners enjoy crafting accessories from old toys, kandi and perler beads, weaving, braiding, crocheting novelty yarn and ribbon, and designing and printing their own textiles. In addition to this act of making, decora practitioners also incorporate purchases from specialty brands like 6%DOKI DOKI, Nile Perch, ACDC Rag, YOSUKE USA, and minacute. According to our interviewees, whom we consulted in 2020, excess is key; as Momo told us: “if it’s too plain, it’s not decora”. Decora uses clashing, vibrant, electric colours, and a wild variety of kawaii versions of monsters, characters, and food which appear as motifs on their clothing (Groom 193; Yagi 17). Clashing textures and items—such as a sweat jackets, gauzy tutus, and plastic toy tiaras—are also a key concept (Koga 81). Colour is extended to practitioners’ hair through colourful hair dyes, and the application of stickers, bandaids, and jewels across their cheeks and nose (Rose, Kurebayashi and Saionji). These principles are illustrated in fig. 1, a street snap from 2015 of our co-author, Kurebayashi. Working with the contrasting primary colours across her hair, clothes, and accessories, she incorporates both her own handmade garments and found accessories to form a balanced outfit. Her Lisa Frank cat purse, made from a psychedelic vibrant pink faux fur, acts as a salient point to draw in our eyes to a cacophony of colour throughout her ensemble. The purse is a prized item from her own collection that was a rare find on Mercari, an online Japanese auction Website, 15 years ago. Her sweater dress is handmade, with a textile print she designed herself. The stickers on the print feature smiley faces, rainbows, ducks, and candy—all cheap and cheerful offerings from a discount store. Through intense layering and repetition, Kurebayashi has created a collage that is reminiscent of the clips and bracelets that decorate her hair and wrists. This collage also represents the colour, fun, and whimsy that she immerses herself in everyday. Her platform shoes are by Buffalo London, another rare find for her collection. Her hair braids are handmade by Midoroya, an online artist, which she incorporates to create variety in the textures in her outfit from head to toe. Peeking beneath her sweater is a short colourful tutu that floats and bounces with each step. Together the items converge and sing, visually loud and popping against the urban landscape. Fig. 1: Kurebayashi’s street snap in an decora fashion outfit of her own styling and making, 2015. Given the street-level nature of decora fashion, stories of its origins draw on oral histories of practitioners, alongside writings from designers and stores that cater to this group (Ash). Its emergence was relatively organic in the early 1990s, with groups enjoying mixing and combining found objects and mis-matching clothing items. Initially, decorative styles documented in street photography used a dark colour palette with layers of handmade accessories, clips, and decorations, and a Visual-kei influence. Designers such as Sebastian Masuda, who entered the scene in 1995, also played a key role by introducing accessories and clothes inspired by vintage American toys, Showa era (1926-1989) packaging, and American West Club dance culture (Sekikawa and Kumagi 22–23). Pop idols such as Tomoe Shinohara and Kyary Pamyu Pamyu are also key figures that have contributed to the pop aesthetic of decora. While decora was already practiced prior to the release of Shinohara’s 1995 single Chaimu, her styling resonated with practitioners and motivated them to pursue a more “pop” aesthetic with an emphasis on bright colours, round shapes, and handmade colourful accessories. Shinohara herself encouraged fans to take on a rebelliously playful outlook and presentation of self (Nakao 15–16; Kondō). This history resonates with more recent pop idol Kyary Pamyu Pamyu’s costuming and set design, which was directed by Sebastian Masuda. Kyary’s kawaii fashion preceded her career, as she regularly participated in the Harajuku scene and agreed to street snaps. While the costuming and set design for her music videos, such as Pon Pon Pon, resonate with the Harajuku aesthetic, her playful persona diverges. Her performance uses humour, absurdity, and imperfection to convey cuteness and provide entertainment (Iseri 158), but practitioners in Harajuku do not try to replicate this performance; Shinohara and Kyary’s stage persona promotes ‘immaturity’ and ‘imperfection’ as part of their youthful teenage rebellion (Iseri 159), while kawaii fashion practitioners prefer not to be seen in this light. When considering the toys, stickers, and accessories incorporated into decora fashion, and the performances of Shinohara and Kyary, it is understandable that some outsiders may interpret the fashion as a desire to return to childhood. Some studies of kawaii fashion more broadly have interpreted the wearing of clothing like this as a resistance to adulthood and infantile regression (e.g., Kinsella 221–222; Winge; Lunning). These studies suggest that practitioners desire to remain immature in order to “undermin[e] current ideologies of gender and power” (Hasegawa 140). In particular, Kinsella in her 1995 chapter “in Japan” asserts that fashion like this is an attempt to act “vulnerable in order to emphasize … immaturity and inability to carry out social responsibilities” (241), and suggests that this regression is “self-mutilation [which denies] the existence of a wealth of insights, feelings and humour that maturity brings with it” (235). This view has spread widely in writing about kawaii fashion, and Steele, Mears, Kawamura, and Narumi observe for instance that “prolonging childhood is compelling” as an attractive component of Harajuku culture (48). While we recognise that this literature uses the concept of “childishness” to acknowledge the rebellious nature of Harajuku fashion, our participants would like to discourage this interpretation of their practice. In particular, participants highlighted their commitment to studies, paying bills, caring for family members, and other markers they felt indicated maturity and responsibility. They also found this belief that they wanted to deny themselves adult “insights, feelings and humour” deeply offensive as it disregards their lived experience and practice. From a Sociological perspective, this infantilising interpretation is concerning as it reproduces Orientalist framings of Japanese women who enjoy kawaii culture as dependent and submissive, rather than savvy consumers (Bow 66–73; Kalnay 95). Furthermore, this commentary on youth cultures globally, which points to an infantilisation of adulthood (Hayward 230), has also been interrogated by scholars as an oversimplistic reading that doesn’t recognise the rich experiences of adults who engage in these spaces while meeting milestones and responsibilities (Woodman and Wyn; Hodkinson and Bennett; Bennett). Through our lived experience and work with the decora fashion community, we offer in this essay an alternative account of what kawaii means to these practitioners. We believe that agency, energy, and vibrancy is central to the practice of decora fashion. Rather than intending to be immature, practitioners are looking for vibrant ways to exist. A New Materialist lens offers a framework with which we can consider this experience. For example, our informant Momota, in rejecting the view that her fashion was about returning to childhood, explained that decora fashion was “rejuvenating” because it gave them “energy and power”. Elizabeth Groscz in her essay on freedom in New Materialism encourages us to consider new ways of living, not as an expression of “freedom from” social norms, but rather “freedom to” new ways of being, as expression of their “capacity for action” (140). In other words, rather than seeking freedom from adult responsibilities and regressing into a state where one is unable to care for oneself, decora fashion is a celebration of what practitioners are “capable of doing” (Groscz 140–141) by finding pleasure in collecting and making. Through encounters with kawaii objects, and the act of creating through these materials, decora fashion practitioners’ agential capacities are increased through experiences of elation, excitement and pleasure. Colourful Treasures, Fluttering Hearts: The Pleasures of Collecting kawaii Matter Christine Yano describes kawaii as having the potential to “transform the mundane material world into one occupied everywhere by the sensate and the sociable” (“Reach Out”, 23). We believe that this conceptualisation of kawaii has strong links to New Materialist theory. New Materialism highlights the ways in which human subjects are “are unstable and emergent knowing, sensing, embodied, affective assemblages of matter, thought, and language, part of and inseparable from more-than human worlds” (Lupton). Matter in this context is a social actor in its own right, energising and compelling practitioners to incorporate them into their everyday lives. For example, kawaii matter can move us to be more playful, creative, and caring (Aiwaza and Ohno; Nishimura; Yano, Pink Globalization), or help us relax and feel calm when experiencing high levels of stress (Stevens; Allison; Yano, “Reach Out”). Studies in the behavioral sciences have shown how kawaii objects pique our interest, make us feel happy and excited, and through sharing our excitement for kawaii things become kinder and more thoughtful towards each other (Nittono; Ihara and Nittono; Kanai and Nittono). Decora fashion practitioners are sensitive to this sensate and sociable aspect of kawaii; specific things redolent with “thing-power” (Bennett) shine and twinkle amongst the cultural landscape and compel practitioners to gather them up and create unique outfits. Decora fashion relies on an ongoing hunt for objects to upcycle into fashion accessories, thrifting second-hand goods in vintage stores, dollar stores, and craft shops such as DAISO, Omocha Spiral, and ACDC Rag. Practitioners select plastic goods with smooth forms and shapes, and soft, breathable, and light clothing, all with highly saturated colours. Balancing the contrast of colours, practitioners create a rainbow of matter from which they assemble their outfits. The concept of the rainbow is significant to practitioners as the synergy of contrasting colours expresses its own kawaii vitality. As our interviewee, Kanepi, described, “price too can be kawaii” (Yano, Pink Globalization 71); affordable products such as capsule toys and accessories allow practitioners to amass large collections of glistening and twinkling objects. Rare items are also prized, such as vintage toys and goods imported from America, resonating with their own “uniqueness”, and providing a point of difference to the Japanese kawaii cultural landscape. In addition to the key principles of colour, rarity, and affordability, there is also a personalised aspect to decora fashion. Amongst the mundane racks of clothing, toys, and stationary, specific matter twinkles at practitioners like treasures, triggering a moment of thrilling encounter. Our interviewee Pajorina described this moment as having a “fateful energy to it”. All practitioners described this experience as “tokimeki” (literally, a fluttering heart beat), which is used to refer to an experience of excitement in anticipation of something, or the elating feeling of infatuation (Occhi). Our interviewees sought to differentiate this experience of kawaii from feelings of care towards an animal or children through writing systems. While the kanji for “kawaii” was used to refer to children and small animals, the majority of participants wrote “kawaii” to express the vivid and energetic qualities of their fashion. We found each practitioner had a tokimeki response to certain items that and informed their collecting work. While some items fit a more mainstream interpretation of kawaii, such as characters like Hello Kitty, ribbons, and glitter, other practitioners were drawn to non-typical forms they believed were kawaii, such as frogs, snails, aliens, and monsters. As our interviewee Harukyu described: “I think people’s sense of kawaii comes from different sensibilities and perspectives. It’s a matter of feelings. If you think it is kawaii, then it is”. Guided by individual experiences of objects on the shop shelves, practitioners select things that resonate with their own inner beliefs, interests, and fantasies of what kawaii is. In this regard, kawaii matter is not “structured” or “fixed” but rather “emergent through relations” that unfold between the practitioner and the items that catch their eye in a given moment (Thorpe 12). This offers not only an affirming experience through the act of creating, but a playful outlet as well. By choosing unconventional kawaii motifs to include in their collection, and using more standard kawaii beads, jewels, and ribbons to enhance the objects’ cuteness, decora fashion practitioners are transforming, warping, and shifting kawaii aesthetic boundaries in new and experimental ways (Iseri 148; Miller 24–25). As such, this act of collecting is a joyous and elating experience of gathering and accumulating. Making, Meaning, and Memory: Creating kawaii Assemblages Once kawaii items are amassed through the process of collecting, their cuteness is intensified through hand-making items and assembling outfits. One of our interviewees, Momo, explained to us that this expressive act was key to the personalisation of their clothes as it allows them to “put together the things you like” and “incorporate your own feelings”. For example, the bracelets in fig. 2 are an assemblage made by our co-author Kurebayashi, using precious items she has collected for 10 years. Each charm has its own meaning in its aesthetics, memories it evokes, and the places in which it was found. Three yellow rubber duck charms bob along strands of twinkling pink and blue bubble-like beads. These ducks, found in a bead shop wholesaler while travelling in Hong Kong, evoke for Kurebayashi an experience of a bubble bath, where one can relax and luxuriate in self care. Their contrast with the pink and blue—forming the trifecta of primary colours—enhances the vibrant intensity of the bracelet. A large blue bear charm, contrasting in scale and colour, swings at her wrist, its round forms evoking Lorenz’s Kindchenschema. This bear charm is another rare find from America, a crowning jewel in Kurebayashi’s collection. It represents Kurebayashi’s interest in fun and colourful animals as characters, and as potential kawaii friends. Its translucent plastic form catches the light as it glistens. To balance the colour scheme of her creation, Kurebayashi added a large strawberry charm, found for just 50 Yen in a discount store in Japan. Together these objects resonate with key decora principles: personal significance, rarity, affordability, and bright contrasting colours. While the bear and duck reference childhood toys, they do not signify to Kurebayashi a desire to return to childhood. Rather, their rounded forms evoke a playful outlook on life informed by self care and creativity (Ngai 841; Rose). Through bringing the collection of items together in making these bracelets, the accessories form an entanglement of kawaii matter that carries both aesthetic and personal meaning, charged with memories, traces of past travels, and a shining shimmering vitality of colour and light. Fig. 2: Handmade decora fashion bracelet by Kurebayashi, 2022. The creation of decora outfits is the final act of expression and freedom. In this moment, decora fashion practitioners experience elation as they gleefully mix and match items from their collection to create their fashion style. This entanglement of practitioner and kawaii matter evokes what Gorscz would describe as “free acts … generated through the encounter of life with matter” (151). If we return to fig. 1, we can see how Kurebayashi and her fashion mutually energise each other as an expression of colourful freedom. While the objects themselves are found through encounters and given new life by Kurebayashi as fashion items, they also provide Kurebayashi with tools of expression that “expand the variety of activities” afforded to adults (Gorscz 154). She feels elated, full of feeling, insight, and humour in these clothes, celebrating all the things she loves that are bright, colourful, and fun. Conclusion In this essay, we have used New Materialist theory to illustrate some of the ways in which kawaii matter energises decora fashion practitioners, as an expression of what Gorscz would describe as “capacity for action” and a “freedom towards” new modes of expression. Practitioners are sensitive to kawaii’s affective potential, motivating them to search for and collect items that elate and excite them, triggering moments of thrilling encounters amongst the mundanity of the stores they search through. Through the act of making and assembling these items, practitioners form an entanglement of matter charged with their feelings, memories, and the vitality and vibrancy of their collections. Like shining rainbows in the streets, they shimmer and shine with kawaii life, vibrancy, and vitality. Acknowledgements This article was produced with the support of a Vitalities Lab Scholarship, UNSW Sydney, a National Library of Australia Asia Studies scholarship, as well as in-kind support from the University of Tokyo and the Japan Foundation Sydney. 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La combinatoire extrémale est l'une des branches les plus vigoureuses des mathématiques combinatoires au cours des dernières décennies, et elle a été largement utilisée en informatique, en conception de réseaux et en conception de codage. Elle se concentre sur la détermination de la taille maximale ou minimale possible de certaines structures combinatoires, sous certaines conditions ou contraintes. Les ensembles hôtes peuvent être des graphes, des digraphes, des graphes aléatoires, des hypergraphes, des entiers, des nombres premiers, des ensembles, des graphes avec arêtes colorées, etc. Les structures locales peuvent être des appariements, des cliques, des cycles, des arbres, des sous-graphes couvrants (facteurs F, cycles Hamiltoniens), des familles d'intersection, des progressions arithmétiques, des solutions pour certaines équations (par exemple, x+y=z), des sous-graphes arc-en-ciel, etc. En particulier, la théorie des graphes extrémaux est une branche importante de la combinatoire extrémale, qui traite principalement de la manière dont les propriétés générales d'un graphe contrôlent la structure locale du graphe. Nous étudions l'existence d'un cycle Hamiltonien rainbow dans les systèmes de k-graphes, l'existence d'un appariement parfait rainbow dans les systèmes de k-graphes et l'existence d'un cycle long arc-en-ciel dans des graphes correctement colorés par les arêtes
Extremal Combinatorics is one of the most vigorous branch of Combinatorial Mathematics in recent decades and it has been widely used in Computer Science, Network Design and Coding Design. It focuses on determining the maximum or minimum possible size of certain combinatorial structures, subject to certain conditions or constraints. The host sets could be graphs, digraphs, random graphs, hypergraphs, integers, primes, sets, edge-colored graphs and so on. The local structures could be matchings, cliques, cycles, trees, spanning subgraphs (F-factors, Hamilton cycles), intersecting families, arithmetic progressions, solutions for some equations (e.g. x₊y₌z), rainbow subgraphs and so on. In particular, Extremal Graph Theory is a significant branch of Extremal Combinatorics, which primarily explores how the overall properties of a graph influence its local structures. We study the existence of a rainbow Hamilton cycle in k-graph systems, the existence of rainbow perfect matching in k-graph systems, and the existence of long rainbow cycle in properly edge-colored graphs