University of IsfahanTransactions on Combinatorics2251-865714420251201Star-critical connected Ramsey numbers for 2-colorings of complete graphs2112222865110.22108/toc.2024.140839.2157ENMonuMounDepartment of Mathematics, Central University of Haryana, Haryana, IndiaJagjeetJakharDepartment of Mathematics, Central University of Haryana, Haryana, India.MarkBuddenDepartment of Mathematics and Computer Science, Western Carolina University, Cullowhee, NC 28723Journal Article20240228This paper builds upon Sumner's work by further investigating the concept of connected Ramsey numbers, specifically focusing on star-critical connected Ramsey numbers. We obtain star-critical connected Ramsey numbers for several cases of trees versus complete graphs, stars versus stars, and paths versus paths. The connected Ramsey number for a star versus $K_3$ is also evaluated. Exact values are also obtained for the connected Ramsey numbers of $K_{1,n}$ versus $K_3$. This research explores the interplay between connectivity and graph coloring within the context of Ramsey theory.https://toc.ui.ac.ir/article_28651_1d475a3047b6054f8fd40fc12ec44034.pdfUniversity of IsfahanTransactions on Combinatorics2251-865714420251201The orders of subgroup products and coset products2232502867010.22108/toc.2024.141562.2176ENShigeruTakamuraDepartment of Mathematics, Faculty of Science, Kyoto University, Kyoto, JAPANJournal Article20240521A <em>sect </em>is a subset of a group given by the product of a finite number of subgroups. It is generally not a direct product nor even a subgroup of the group. For finite groups, the orders of sects are their basic invariants. In this paper we describe properties of the orders of sects, such as divisibility and inequalities, which give constraints on the possible values of the orders of sects. We further consider <em>clans</em>, which are subsets of groups given by products of finite numbers of cosets. We also describe properties of their orders.https://toc.ui.ac.ir/article_28670_ebc3ac219eb683b98452f6d44ee439b0.pdfUniversity of IsfahanTransactions on Combinatorics2251-865714420241115Three new classes of binomial Fibonacci sums2512592876310.22108/toc.2024.141371.2171ENRobertFrontczakIndependent Researcher, 72762 Reutlingen, GermanyJournal Article20240504In this paper, we introduce three new classes of binomial sums involving Fibonacci (Lucas) numbers and weighted binomial coefficients. One particular result is linked to a problem proposal recently published in the journal The Fibonacci Quarterly.https://toc.ui.ac.ir/article_28763_a471f51243361ab8ec7c8bb8c280b4f7.pdfUniversity of IsfahanTransactions on Combinatorics2251-865714420241117On twin EP numbers2612702878110.22108/toc.2024.138412.2092ENÖmerEğecioğluDepartment of Computer Science, University of California Santa Barbara, CA 93106, USABünyaminŞahinDepartment of Mathematics, Faculty of Science, Selçuk University, Konya 42130, TurkeyJournal Article20230716EP numbers were introduced by Estrada and Pogliani in 2008. These are positive integers $E(n)$ defined as the product of $n$ and the sum of the digits of $n$. Estrada and Pogliani suspected that there may be infinitely many twin EP numbers; i.e. those pairs in this sequence that differ by one. It is relatively easy to show that three consecutive EP numbers do not exist, and that no pair $E(n)$ and $E(m)$ can be twins for infinitely many bases $b$.
The main contribution of our work is the result that indeed there are infinitely many twin EP numbers over any base.
The proof is constructive and makes use of elementary properties of natural numbers. The forms of the twin EP numbers presented are derived from continued fractions. The behavior of the series of the reciprocals of twin EP numbers is also considered.https://toc.ui.ac.ir/article_28781_dc93c3e628e7d0c7006fe39e596c0e12.pdfUniversity of IsfahanTransactions on Combinatorics2251-865714420241117$G$-designs for the connected triangular bicyclic graphs with nine edges2712812894410.22108/toc.2024.140831.2156ENBryanFreybergDepartment of Mathematics and Statistics, University of Minnesota Duluth, Duluth, MNDaliborFroncekDepartment of Mathematics and Statistics
University of Minnesota Duluth
1117 University Dr.
Duluth, MN 55812-3000
USA0000-0003-4528-2059JoelJeffriesDepartment of Mathematics, Iowa State University, 411 Morrill Road, Ames, USAGrettaJensenDepartment of Mathematics and Statistics, University of Minnesota Duluth, 1117 University Drive, Duluth, USAAndrewSailstadSchool of Mathematics, University of Minnesota, 127 Vincent Hall 206 Church St. SE, Minneapolis, USAJournal Article20240227A $G$-design of order $n$ is a decomposition of the complete graph $K_n$ into isomorphic copies of $G$. We show that if $G$ is a connected bicyclic graph with nine edges containing two triangles, a $G$-design of order $n$ exists whenever $n \equiv 0,1 \pmod{18}$.https://toc.ui.ac.ir/article_28944_0251192f0ac271abed0c719b103dc4d4.pdf