REDUCTION OF THE BERGE-FULKERSON CONJECTURE TO CYCLICALLY 5-EDGE-CONNECTED SNARKS

The Berge-Fulkerson conjecture, originally formulated in the language of mathematical programming, asserts that the edges of every bridgeless cubic (3-valent) graph can be covered with six perfect matchings in such a way that every edge is in exactly two of them. As with several other classical conjectures in graph theory, every counterexample to the Berge-Fulkerson conjecture must be a non-3-edge-colorable cubic graph. In contrast to Tutte's 5-flow conjecture and the cycle double conjecture, no nontrivial reduction is known for the Berge-Fulkerson conjecture. In the present paper, we prove that a possible minimum counterexample to the conjecture must be cyclically 5-edge-connected

Research on elisions in preschool age children

The key activities in child’s phonemic awareness development, are those which require manipulation with phonemes. One of the possibilities for work on the phoneme level and train the ability to manipulate with phonemes are elisions. These tasks assume the children\u27s skills to isolate phonemes (or larger units) and then omit them. The aim is to identify the remaining part of the word that was created by omitting the corresponding phoneme, syllable or parts of the word. This theoretical-empirical study presents some theoretical issues connected with the topic, and it focuses on the results acquired from children in the preschool age in Slovakia. The research was conducted with 866 children at the age of four to seven years. It was focused on the ability to realize phoneme elision, i.e. to isolate the sound in a word and afterward pronounce the word which arises from omitting a certain sound. Testing the phoneme elision has shown that it is the most demanding phonemic ability for children taking part in the research. The overall success in all age categories was only 23%. The test revealed children with above the average developed ability to the established norm. The results are part of a more extensive research which is focused on the development of a complex tool used to evaluate the level of phonemic awareness. The paper is the outcome of the VEGA project no. 1/0637/16 entitled The Development of the Diagnostic Instrument for the Assessment of the Level of Phonemic Awareness of Preschool Age Children.As atividades-chave no desenvolvimento da consciência fonética da criança são as que requerem manipulação com os fonemas. Uma das possibilidades de trabalhar no nível dos fonemas e treinar a capacidade de manipulação dos fonemas são as elisões. Essas tarefas pressupõem as capacidades das crianças para isolarem fonemas (ou unidades maiores) e, em seguida, omiti-los. O objetivo é identificar o resto da palavra que foi criada, omitindo o fonema, sílaba ou partes da palavra correspondente. Este estudo teórico-empírico apresenta algumas questões teóricas relacionadas com o tema e centra-se nos resultados obtidos com crianças em idade pré-escolar na Eslováquia. A pesquisa foi realizada com 866 crianças na idade de 4 a 7 anos. Focou-se na capacidade de perceber a elisão de fonemas, ou seja, isolar o som em uma palavra e, depois, pronunciar a palavra que surge da omissão de um determinado som. O teste de elisão fonética mostrou que é a competência fonética mais exigente para as crianças participantes no estudo. O sucesso geral em todas as categorias de idade foi de apenas 23%. O teste revelou crianças com capacidade desenvolvida acima da média para a norma estabelecida. Os resultados são parte de uma pesquisa mais ampla que tem como foco o desenvolvimento de um complexo instrumento de avaliação do nível de consciência fonética. O trabalho é resultado do projeto VEGA nº. 1/0637/16 intitulado “O Desenvolvimento do Instrumento Diagnóstico para Avaliação do Nível de Consciência Fonética de Crianças em Idade Pré-escolar”

Perfect-matching covers of cubic graphs with colouring defect 3

The colouring defect of a cubic graph is the smallest number of edges left uncovered by any set of three perfect matchings. While $3$-edge-colourable graphs have defect $0$, those that cannot be $3$-edge-coloured have defect at least $3$. We show that every bridgeless cubic graph with defect $3$ can have its edges covered with at most five perfect matchings, which verifies a long-standing conjecture of Berge for this class of graphs. Moreover, we determine the extremal graphs

On the Frank number and nowhere-zero flows on graphs

An edge e of a graph G is called deletable for some orientation o if the restriction of o to G - e is a strong orientation. In 2021, Horsch and Szigeti proposed a new parameter for 3-edge-connected graphs, called the Frank number, which refines k-edge-connectivity. The Frank number is defined as the minimum number of orientations of G for which every edge of G is deletable in at least one of them. They showed that every 3-edge-connected graph has Frank number at most 7 and that in case these graphs are also 3-edge-colourable graphs the parameter is at most 3. Here we strengthen the latter result by showing that such graphs have Frank number 2, which also confirms a conjecture by Bar ' at and Bl ' azsik. Furthermore, we prove two sufficient conditions for cubic graphs to have Frank number 2 and use them in an algorithm to computationally show that the Petersen graph is the only cyclically 4-edge-connected cubic graph up to 36 vertices having Frank number greater than 2

Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44

status: publishe