Comb Theory Research Articles

Multi-part cross-intersecting families

Let A⊆[n]a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}\\subseteq {[n]\\atopwithdelims ()a}$$\\end{document} and B⊆[n]b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {B}}\\subseteq {[n]\\atopwithdelims ()b}$$\\end{document} be two families of subsets of [n], we say A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}$$\\end{document} and B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {B}}$$\\end{document} are cross-intersecting if A∩B≠∅\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\cap B\ e \\emptyset $$\\end{document} for all A∈A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\in {\\mathcal {A}}$$\\end{document}, B∈B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B\\in {\\mathcal {B}}$$\\end{document}. In this paper, we study cross-intersecting families in the multi-part setting. By characterizing the independent sets of vertex-transitive graphs and their direct products, we determine the sizes and structures of maximum-sized multi-part cross-intersecting families. This generalizes the results of Hilton’s (J Lond Math Soc 15(2):369–376, 1977) and Frankl–Tohushige’s (J Comb Theory Ser A 61(1):87–97, 1992) on cross-intersecting families in the single-part setting.

Nurses’ engagement in antimicrobial stewardship and its influencing factors: A cross-sectional study

ObjectiveThis study aimed to investigate the level and influencing factors of nurses’ antimicrobial stewardship (AMS) engagement in China based on the capability, opportunity, motivation, and behavior (COM-B) theory, providing valuable insights for developing effective strategies to improve nursing quality in AMS. MethodsThis cross-sectional study was conducted in 17 tertiary hospitals in Hunan, China, from November 2021 to January 2022. A total of 4,514 nurses were selected. The Nurse AMS Engagement Questionnaire (NAEQ), developed using the COM-B theory, was used for evaluation. The questionnaire included capability (14 items), opportunity (7 items), motivation (6 items), and behavior (12 items) four dimensions, 39 items. ResultsThe total NAEQ score was 155.08 ± 27.12, indicating a moderate level. The score of the capability, opportunity, motivation, and behavior dimensions were 52.33 ± 13.48, 28.64 ± 5.76, 24.57 ± 4.57 and 49.53 ± 8.83, respectively. Significant differences in nurses’ AMS engagement were based on professional titles, whether working as a part-time infection control nurse, whether knowing the AMS teams and the defined daily doses of antibiotics, department type, the deployment of clinical pharmacists, and frequency of antimicrobial training and physician-nurse joint rounds (P < 0.05). Nurses with junior titles had higher scores on the NAEQ than nurses with intermediate titles (P < 0.05). Nurses who worked as part-time infection control nurses, knew the AMS team, and the defined daily doses of antibiotics had higher NAEQ scores than those who didn’t (P < 0.01). Nurses working in the ICU and infectious disease department had lower NAEQ scores than those in other departments, such as the ear, nose, and throat (ENT) department (P < 0.01). Nurses who had clinical pharmacists deployed in their department had higher NAEQ scores than those without or unclear deployment (P < 0.01). Furthermore, nurses who received more frequent antimicrobial training and participated in physician-nurse joint rounds had higher NAEQ scores (P < 0.01). ConclusionMultiple strategies, including enhanced education and training and improved multidisciplinary communication and collaboration, are expected to improve nurse AMS engagement. It is important to give more attention to nurses with intermediate professional titles, less experience, and those working in specific departments.

Improving personalised care, through the development of a service evaluation tool to assess, understand and monitor delivery

Systematically implementing personalised care has far reaching benefits to individuals, communities and health and social care systems. If done well, personalised care can result in better health outcomes and experiences,...

Women's experience of polycystic ovary syndrome management: A systematic review and meta-synthesis.

Polycystic ovary syndrome (PCOS) is a common chronic condition in women of child-bearing age. There is currently no effective treatment, so early and long-term management is essential. However, there are many problems in the practice of disease management in women with PCOS that make it difficult to achieve good outcomes. To explore women's experience of PCOS management and identify the relevant facilitators and barriers to management. A structured search was undertaken in five bibliographic databases (MEDLINE, Web of Science, CINAHL, Embase, PubMed, and Cochrane) from the date of establishment of the database up to December 2022. All qualitative and mixed-methods studies available in English describing the experience of PCOS management from the patients' perspective were included. The Joanna Briggs Institute Qualitative Assessment and Review Instrument was used to appraise study quality. The evidence was synthesized using a pragmatic meta-aggregative approach guided by the capability, opportunity, and motivation model of behavior (COM-B). A total of 13 studies were included with 85 equivocal findings and 12 credible findings. The findings were meta-aggregated into three themes: (1) capability of women with PCOS, including patients' attitudes toward disease and management, knowledge, and skills of the disease; (2) opportunities in PCOS management, including information about PCOS, diagnostic delay, disease characteristics, disease management plan, and logistical and environmental problems; and (3) motivation in PCOS management, including impact of symptoms, perceived needs, support and feedback, and unpleasant medical experience. This study identifies facilitators and barriers to PCOS management from the patient perspective, which can guide the design and implementation of PCOS management programs for patients. This study also provides information for future research into how the COM-B theory can be incorporated into specific management plans to promote patient action.

Graph isomorphism: physical resources, optimization models, and algebraic characterizations

In the (G, H)-isomorphism game, a verifier interacts with two non-communicating players (called provers), by privately sending each of them a random vertex from either G or H. The goal of the players is to convince the verifier that the graphs G and H are isomorphic. In recent work along with Atserias et al. (J Comb Theory Ser B 136:89–328, 2019) we showed that a verifier can be convinced that two non-isomorphic graphs are isomorphic, if the provers are allowed to share quantum resources. In this paper we model classical and quantum graph isomorphism by linear constraints over certain complicated convex cones, which we then relax to a pair of tractable convex models (semidefinite programs). Our main result is a complete algebraic characterization of the corresponding equivalence relations on graphs in terms of appropriate matrix algebras. Our techniques are an interesting mix of algebra, combinatorics, optimization, and quantum information.

On monohedral tilings of a regular polygon

A tiling of a topological disc by topological discs is called monohedral if all tiles are congruent. Maltby (J Comb Theory Ser A 66:40–52, 1994) characterized the monohedral tilings of a square by three topological discs. Kurusa et al. (Mediterr J Math 17:156, 2020) characterized the monohedral tilings of a circular disc by three topological discs. The aim of this note is to connect these two results by characterizing the monohedral tilings of any regular n-gon with at most three tiles for any n≥5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 5$$\\end{document}.

The Asymptotic Number of Score Sequences

A tournament on a graph is an orientation of its edges. The score sequence lists the in-degrees in non-decreasing order. Works by Winston and Kleitman (J Comb Theory Ser A 35(2):208–230, 1983) and Kim and Pittel (J Comb Theory Ser A 92(2):197–206, 2000) showed that the number S_n of score sequences on the complete graph K_n satisfies S_n=Theta (4^n/n^{5/2}). By combining a recent recurrence relation for S_n in terms of the Erdős–Ginzburg–Ziv numbers N_n with the limit theory for discrete infinitely divisible distributions, we observe that n^{5/2}S_n/4^nrightarrow e^lambda /2sqrt{pi }, where lambda =sum _{k=1}^infty N_k/k4^k. This limit agrees numerically with the asymptotics of S_n conjectured by Takács (J Stat Plan Inference 14(1):123–142, 1986). We also identify the asymptotic number of strong score sequences, and show that the number of irreducible subscores in a random score sequence converges in distribution to a shifted negative binomial with parameters r=2 and p=e^{-lambda }.

A direct proof of the non-existence of a MOL(6)

It is well known that a Latin Square of order 6 has no orthogonal mate. Many proofs are known, some of which are very short, see for instance (Betten in Unterricht 36:449–453, 1983; Beth et al. in Design theory, Bibliographisches Institut Mannheim Wien, Zürich, 1985; Tarry in Comptes Rendus Ass Franc Sci Nat 1900(2), 170–203, 1901; Stinson in J Comb Theory A 36:373–376, 1984). This paper provides a short proof of this fact which avoids a case distinction on the isomorphism types of the Latin Square. We observe that any Latin Square of order 6 falls within exactly one of three categories. Either it has two rows which form a permutation whose cycle type is three transpositions. Or it has no subsquare of order 2, or it is a single Latin Square with symmetry group the rotation group of the cube of order 24. In each case, the nonexistence of an orthogonal mate can be seen quickly.

From projects to systems: Being explicit about assumptions in systems-level theories of change

Projects and programs have long been the primary instrument for achieving development goals in low- and middle-income countries. One criticism of the project-focused approach is its failure to focus on broader system-level changes. This paper explores how Mayne's COM-B Theory of Change model can enhance the evaluation of how projects and system-level investments can lead to system-level changes, especially in a development context. Using a real-world example, we offer several evaluation questions to initiate thinking about what might be needed to extend the ideas in the COM-B theory of change to facilitate better interrogation of systems-level change efforts.

Sub-exponential Time Parameterized Algorithms for Graph Layout Problems on Digraphs with Bounded Independence Number

Fradkin and Seymour (J Comb Theory Ser B 110:19–46, 2015) defined the class of digraphs of bounded independence number as a generalization of the class of tournaments. They argued that the class of digraphs of bounded independence number is structured enough to be exploited algorithmically. In this paper, we further strengthen this belief by showing that several cut problems that admit sub-exponential time parameterized algorithms (a trait uncommon to parameterized algorithms) on tournaments, including Directed Feedback Arc Set, Directed Cutwidth and Optimal Linear Arrangement, also admit such algorithms on digraphs of bounded independence number. Towards this, we rely on the generic approach of Fomin and Pilipczuk (in: Proceedings of the Algorithms—ESA 2013—21st Annual European Symposium, Sophia Antipolis, France, September 2–4, 2013, pp. 505–516, 2013), where to get the desired algorithms, it is enough to bound the number of k-cuts in digraphs of bounded independence number by a sub-exponential FPT function (Fomin and Pilipczuk bounded the number of k-cuts in transitive tournaments). Specifically, our main technical contribution is a combinatorial result that proves that the yes-instances of the problems (defined above) have a sub-exponential number of k-cuts. We prove this bound by using a combination of chromatic coding, inductive reasoning and exploiting the structural properties of these digraphs.

FAKTOR YANG MENDASARI PERILAKU KEPATUHAN MAHASISWA DI YOGYAKARTA TERHADAP PROTOKOL KESEHATAN COVID-19 PENDEKATAN COM-B THEORY OF CHANGE

Background: The COVID-19 pandemic has had an impact on the world of education, that were very limited. Nowadays, with the current decline in COVID-19 cases in Yogyakarta, teaching and learning activities have started to be carried out offline. It is necessary to be aware that this teaching and learning activity risks the occurrence of clusters of the spread of the COVID-19 virus. Therefore, student compliance with health protocols is very important to be encouraged. The factors that underlie adherence to health protocols need to be studied more in order to maintain and improve student compliance. Research purpose: The research aims to study the factors that underlie student compliance in implementing health protocols. Methods: The research was conducted using a descriptive qualitative method, where the informants were University students in Yogyakarta. The interview guide is made with the framework of the Capacity, Opportunity and Motivation Behavior constructs (COM-B Theory of Change). Result: Of the 20 students interviewed, all informants showed adherence to the health protocol. The factors that underlie student compliance include physical ability, knowledge, availability of equipment, facilities and infrastructure, social support from families and those around them, support from government or campus authority’s policies and regulations, motivation in the form of awareness, intention and sense of responsibility. However, some informants stated that there were physical and social barriers in the form of difficulty using masks during strenuous activities, experiencing irritation due to the use of hand washing soap/ hand sanitizer, difficulty maintaining distance in public places, and difficulty avoiding handshakes with the closest people. Conclusion: Physical and social barriers that occur can be overcome by continuing education intervention strategies to the community and disciplinary supervision of health protocols, especially in public places.

Weakly Distance-Regular Digraphs of One Type of Arcs

In this paper, we classify all commutative weakly distance-regular digraphs of girth g and one type of arcs under the assumption that \(p_{(1,g-1),(1,g-1)}^{(2,g-2)}\ge k_{1,g-1}-2\). In consequence, Yang et al. (J Comb Theory Ser A 160:288–315, 2018, Theorem 1.1) is partially generalized by our result.

The geometric reason for the non-existence of a MOL(6)

The problem of Euler concerning the 36 officers, (Euler, in Leonardi Euleri Opera Ser I 7:291–392, 1782), was first solved by Tarry (Comptes rendus Ass Franc Sci Nat 1 (1900) 2:170–203, 1901). Short proofs for the non-existence were given in Betten (Unterricht 36:449–453, 1983), Beth et al. (Design Theory, Bibl. Inst. Mannheim, Wien, Zürich, 1985), Stinson (J Comb Theory A 36:373–376, 1984). This problem is equivalent to the existence of a MOL(6), i. e., a pair of mutually orthogonal latin squares of order 6. Therefore in Betten (Mitt Math Ges Hamburg 39:59–76, 2019; Res Math 76:9, 2021; Algebra Geom 62:815–821, 2021) the structure of a (hypothetical) MOL(6) was studied. Now we combine the old proofs and the new studies and filter out a simple way for the proof of non-existence. The aim is not only to give still other short proofs, but to analyse the problem and reveal the geometric reason for the non-existence of a MOL(6)- and the non-solvability of Euler’s problem.

The complexity of subtree intersection representation of chordal graphs and linear time chordal graph generation

It is known that any chordal graph on n vertices can be represented as the intersection of n subtrees in a tree on n nodes (Gavril in J Comb Theory 16:47–56, 1974). This characterization has been recently used to generate random chordal graphs on n vertices by generating n subtrees of a tree on n nodes. The space (and thus time) complexity of an algorithm generating n subtrees of a tree on n nodes is at least the sum of the sizes of the generated subtrees. The determination of this sum was left as an open question in Şeker et al. (Generation of random chordal graphs using subtrees of a tree. arXiv preprint arXiv:1810.13326, 2018). In this paper, we show that the sum of the sizes of n subtrees in a tree on n nodes is \(\varTheta (m\sqrt{n})\). We also show that we can confine ourselves to contraction-minimal subtree intersection representations because they are sufficient to generate every chordal graph with strictly positive probability. Moreover, the sum of the sizes of the subtrees in a contraction-minimal representation is at most \(2m+n\). We use this result to derive the first linear-time random chordal graph generator. Based on contraction-minimal representations, we also derive connectivity-related structural properties of chordal graphs. Besides these theoretical results, we also conduct experiments to study the quality of the chordal graphs generated by our algorithm and compare them to those generated by existing methods from the literature. Our algorithm does not generate chordal graphs uniformly at random, which is a quite challenging open question, irrespective of the time complexity of the generator. However, our experimental study suggests that the generated graphs have a fairly varied structure as indicated by the sizes of maximal cliques. Furthermore, our algorithm is simple to implement and produces graphs with 10,000 vertices and \(4 \times 10^7\) edges in less than one second on a laptop computer.

Tetravalent edge-transitive Cayley graphs of Frobenius groups

In this paper, we give a characterisation for a class of edge-transitive Cayley graphs and provide a method for constructing edge-transitive graphs of valency 4 with arbitrarily large vertex stabiliser. In particular, in the last section, we obtain certain extensions of the results of Li et al. (Tetravalent edge-transitive Cayley graphs with odd number of vertices, J Comb Theory Ser B 96:164–181, 2006) on half-transitive graphs.

Abelian difference sets with the symmetric difference property

A $$(v,k,\lambda )$$ symmetric design is said to have the symmetric difference property (SDP) if the symmetric difference of any three blocks is either a block or the complement of a block. The designs associated to the symplectic difference sets introduced by Kantor (J Algebra 33:43–58, 1975) have the SDP. Parker (J Comb Theory Ser A 67:23–43, 1994) claimed that the symplectic design on 64 points is the only SDP design on 64 points admitting an abelian regular automorphism group (an abelian difference set). We show in this paper that there is an SDP design on 64 points that is not isomorphic to the symplectic design and yet admits the group $$C_8 \times C_4 \times C_2$$ as a regular automorphism group. This abelian difference set is the first in an infinite family of abelian difference sets whose designs have the SDP and yet are not isomorphic to the symplectic designs of the same order. We define a new method for establishing the non-isomorphism of the two families.

Planar Graphs Without Adjacent Cycles of Length at Most Five are (2, 0, 0)-Colorable

A graph G is $$(d_{1},d_{2},\ldots ,d_{k})$$ -colorable if the vertex set of G can be partitioned into subsets $$V_{1}, V_{2},\ldots , V_{k}$$ such that the subgraph $$G[V_{i}]$$ induced by $$V_{i}$$ has maximum degree at most $$d_{i}$$ for $$i = 1, 2, \ldots , k$$ . Novosibirsk’s Conjecture (Sib lektron Mat Izv 3:428–440, 2006) says that every planar graph without 3-cycles adjacent to cycles of length 3 or 5 is 3-colorable. Borodin et al. (Discrete Math 310: 167–173, 2010) asked whether every planar graph without adjacent cycles of length at most 5 is 3-colorable. Cohen-Addad et al. (J Comb Theory Ser B 122:452–456, 2017) gave a negative answer to both Novosibirsk’s conjecture and Borodin et al.’s problem. Zhang et al. (Discrete Math 339:3032–3042, 2016) asked whether every planar graph without adjacent cycles of length at most 5 is (1, 0, 0)-colorable. In this paper, we show that every planar graph without adjacent cycles of length at most five is (2, 0, 0)-colorable, which improves the result of Chen et al. (Discrete Math 339:886–905, 2016) who proved that every planar graph without cycles of length 4 and 5 is (2, 0, 0)-colorable.

Relationship of two formulations for shortest bibranchings

The shortest bibranching problem is a common generalization of the minimum-weight edge cover problem in bipartite graphs and the minimum-weight arborescence problem in directed graphs. For the shortest bibranching problem, an efficient primal-dual algorithm is given by Keijsper and Pendavingh (J Comb Theory Ser B 73:130–145, 1998), and the tractability of the problem is ascribed to total dual integrality in a linear programming formulation by Schrijver (Ann Discret Math 16:261–280, 1982). Another view on the tractability of this problem is afforded by a valuated matroid intersection formulation by Takazawa (Jpn J Ind Appl Math 29:561–573, 2012). In the present paper, we discuss the relationship between these two formulations for the shortest bibranching problem. We first demonstrate that the valuated matroid intersection formulation can be derived from the linear programming formulation through the Benders decomposition, where integrality is preserved in the decomposition process and the resulting convex programming is endowed with discrete convexity. We then show how a pair of primal and dual optimal solutions of one formulation is constructed from that of the other formulation, thereby providing a connection between polyhedral combinatorics and discrete convex analysis.

Extremal Graphs for Blow-Ups of Keyrings

The blow-up of a graph H is the graph obtained from replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different. Given a graph H and a positive integer n, the extremal number, ex(n, H), is the maximum number of edges in a graph on n vertices that does not contain H as a subgraph. A keyring $$C_s(k)$$ is a $$(k+s)$$ -edge graph obtained from a cycle of length k by appending s leaves to one of its vertices. This paper determines the extremal number and finds the extremal graphs for the blow-ups of keyrings $$C_s(k)$$ ( $$k\ge 3$$ , $$s\ge 1$$ ) when n is sufficiently large. For special cases when $$k=0$$ or $$s=0$$ , the extremal number of the blow-ups of the graph $$C_s(0)$$ (a star) has been determined by Erdos et al. (J Comb Theory Ser B 64:89–100, 1995) and Chen et al. (J Comb Theory Ser B 89: 159–171, 2003), while the extremal number and extremal graphs for the blow-ups of the graph $$C_0(k)$$ (a cycle) when n is sufficiently large has been determined by Liu (Electron J Combin 20: P65, 2013).

Rank-metric codes, linear sets, and their duality

In this paper we investigate connections between linear sets and subspaces of linear maps. We give a geometric interpretation of the results of Sheekey (Adv Math Commun 10:475–488, 2016, Sect. 5) on linear sets on a projective line. We extend this to linear sets in arbitrary dimension, giving the connection between two constructions for linear sets defined in Lunardon (J Comb Theory Ser A 149:1–20, 2017). Finally, we then exploit this connection by using the MacWilliams identities to obtain information about the possible weight distribution of a linear set of rank n on a projective line $$\mathrm {PG}(1,q^n)$$.