{"title":"Asymptotically sharpening the $s$-Hamiltonian index bound","authors":"Sulin Song, Lan Lei, Yehong Shao, H. Lai","doi":"10.46298/dmtcs.8484","DOIUrl":"https://doi.org/10.46298/dmtcs.8484","url":null,"abstract":"For a non-negative integer $sle |V(G)|-3$, a graph $G$ is $s$-Hamiltonian if\u0000the removal of any $kle s$ vertices results in a Hamiltonian graph. Given a\u0000connected simple graph $G$ that is not isomorphic to a path, a cycle, or a\u0000$K_{1,3}$, let $delta(G)$ denote the minimum degree of $G$, let $h_s(G)$\u0000denote the smallest integer $i$ such that the iterated line graph $L^{i}(G)$ is\u0000$s$-Hamiltonian, and let $ell(G)$ denote the length of the longest non-closed\u0000path $P$ in which all internal vertices have degree 2 such that $P$ is not both\u0000of length 2 and in a $K_3$. For a simple graph $G$, we establish better upper\u0000bounds for $h_s(G)$ as follows. begin{equation*} h_s(G)le left{\u0000begin{aligned} & ell(G)+1, &&mbox{ if }delta(G)le 2 mbox{ and }s=0; &\u0000widetilde d(G)+2+lceil lg (s+1)rceil, &&mbox{ if }delta(G)le 2 mbox{\u0000and }sge 1; & 2+leftlceillgfrac{s+1}{delta(G)-2}rightrceil, && mbox{\u0000if } 3ledelta(G)le s+2; & 2, &&{rm otherwise}, end{aligned} right.\u0000end{equation*} where $widetilde d(G)$ is the smallest integer $i$ such that\u0000$delta(L^i(G))ge 3$. Consequently, when $s ge 6$, this new upper bound for\u0000the $s$-hamiltonian index implies that $h_s(G) = o(ell(G)+s+1)$ as $s to\u0000infty$. This sharpens the result, $h_s(G)leell(G)+s+1$, obtained by Zhang et\u0000al. in [Discrete Math., 308 (2008) 4779-4785].","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133600966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Daniel Birmajer, J. Gil, David S. Kenepp, M. Weiner
{"title":"Restricted generating trees for weak orderings","authors":"Daniel Birmajer, J. Gil, David S. Kenepp, M. Weiner","doi":"10.46298/dmtcs.8350","DOIUrl":"https://doi.org/10.46298/dmtcs.8350","url":null,"abstract":"Motivated by the study of pattern avoidance in the context of permutations and ordered partitions, we consider the enumeration of weak-ordering chains obtained as leaves of certain restricted rooted trees. A tree of order $n$ is generated by inserting a new variable into each node at every step. A node becomes a leaf either after $n$ steps or when a certain stopping condition is met. In this paper we focus on conditions of size 2 ($x=y$, $x<y$, or $xle y$) and several conditions of size 3. Some of the cases considered here lead to the study of descent statistics of certain `almost' pattern-avoiding permutations.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128735968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Efficient recurrence for the enumeration of permutations with fixed pinnacle set","authors":"Wenjie Fang","doi":"10.46298/dmtcs.8321","DOIUrl":"https://doi.org/10.46298/dmtcs.8321","url":null,"abstract":"Initiated by Davis, Nelson, Petersen and Tenner (2018), the enumerative study\u0000of pinnacle sets of permutations has attracted a fair amount of attention\u0000recently. In this article, we provide a recurrence that can be used to compute\u0000efficiently the number $|mathfrak{S}_n(P)|$ of permutations of size $n$ with a\u0000given pinnacle set $P$, with arithmetic complexity $O(k^4 + klog n)$ for $P$\u0000of size $k$. A symbolic expression can also be computed in this way for\u0000pinnacle sets of fixed size. A weighted sum $q_n(P)$ of $|mathfrak{S}_n(P)|$\u0000proposed in Davis, Nelson, Petersen and Tenner (2018) seems to have a simple\u0000form, and a conjectural form is given recently by Flaque, Novelli and Thibon\u0000(2021+). We settle the problem by providing and proving an alternative form of\u0000$q_n(P)$, which has a strong combinatorial flavor. We also study admissible\u0000orderings of a given pinnacle set, first considered by Rusu (2020) and\u0000characterized by Rusu and Tenner (2021), and we give an efficient algorithm for\u0000their counting.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126631095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Anna M. Brandenberger, L. Devroye, Marcel K. Goh, Rosie Y. Zhao
{"title":"Leaf multiplicity in a Bienaymé-Galton-Watson tree","authors":"Anna M. Brandenberger, L. Devroye, Marcel K. Goh, Rosie Y. Zhao","doi":"10.46298/dmtcs.7515","DOIUrl":"https://doi.org/10.46298/dmtcs.7515","url":null,"abstract":"This note defines a notion of multiplicity for nodes in a rooted tree and\u0000presents an asymptotic calculation of the maximum multiplicity over all leaves\u0000in a Bienaym'e-Galton-Watson tree with critical offspring distribution $xi$,\u0000conditioned on the tree being of size $n$. In particular, we show that if $S_n$\u0000is the maximum multiplicity in a conditional Bienaym'e-Galton-Watson tree,\u0000then $S_n = Omega(log n)$ asymptotically in probability and under the further\u0000assumption that ${bf E}{2^xi} < infty$, we have $S_n = O(log n)$\u0000asymptotically in probability as well. Explicit formulas are given for the\u0000constants in both bounds. We conclude by discussing links with an alternate\u0000definition of multiplicity that arises in the root-estimation problem.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"142 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127488854","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Marthe Bonamy, Łukasz Bożyk, Andrzej Grzesik, Meike Hatzel, Tomáš Masařík, Jana Novotn'a, Karolina Okrasa
{"title":"Tuza's Conjecture for Threshold Graphs","authors":"Marthe Bonamy, Łukasz Bożyk, Andrzej Grzesik, Meike Hatzel, Tomáš Masařík, Jana Novotn'a, Karolina Okrasa","doi":"10.46298/dmtcs.7660","DOIUrl":"https://doi.org/10.46298/dmtcs.7660","url":null,"abstract":"Tuza famously conjectured in 1981 that in a graph without k+1 edge-disjoint\u0000triangles, it suffices to delete at most 2k edges to obtain a triangle-free\u0000graph. The conjecture holds for graphs with small treewidth or small maximum\u0000average degree, including planar graphs. However, for dense graphs that are\u0000neither cliques nor 4-colorable, only asymptotic results are known. Here, we\u0000confirm the conjecture for threshold graphs, i.e. graphs that are both split\u0000graphs and cographs, and for co-chain graphs with both sides of the same size\u0000divisible by 4.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"88 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115898078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Boolean dimension of a graph and other related parameters","authors":"M. Pouzet, H. S. Kaddour, Bhalchandra D. Thatte","doi":"10.46298/dmtcs.7437","DOIUrl":"https://doi.org/10.46298/dmtcs.7437","url":null,"abstract":"We present the Boolean dimension of a graph, we relate it with the notions of inner, geometric and symplectic dimensions, and with the rank and minrank of a graph. We obtain an exact formula for the Boolean dimension of a tree in terms of a certain star decomposition. We relate the Boolean dimension with the inversion index of a tournament.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"96 3 Pt 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129650710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Márcia R. Cappelle, Erika M. M. Coelho, L. Foulds, Humberto J. Longo
{"title":"Open-independent, open-locating-dominating sets: structural aspects of some classes of graphs","authors":"Márcia R. Cappelle, Erika M. M. Coelho, L. Foulds, Humberto J. Longo","doi":"10.46298/dmtcs.8440","DOIUrl":"https://doi.org/10.46298/dmtcs.8440","url":null,"abstract":"Let $G=(V(G),E(G))$ be a finite simple undirected graph with vertex set\u0000$V(G)$, edge set $E(G)$ and vertex subset $Ssubseteq V(G)$. $S$ is termed\u0000emph{open-dominating} if every vertex of $G$ has at least one neighbor in $S$,\u0000and emph{open-independent, open-locating-dominating} (an $OLD_{oind}$-set for\u0000short) if no two vertices in $G$ have the same set of neighbors in $S$, and\u0000each vertex in $S$ is open-dominated exactly once by $S$. The problem of\u0000deciding whether or not $G$ has an $OLD_{oind}$-set has important applications\u0000that have been reported elsewhere. As the problem is known to be\u0000$mathcal{NP}$-complete, it appears to be notoriously difficult as we show that\u0000its complexity remains the same even for just planar bipartite graphs of\u0000maximum degree five and girth six, and also for planar subcubic graphs of girth\u0000nine. Also, we present characterizations of both $P_4$-tidy graphs and the\u0000complementary prisms of cographs that have an $OLD_{oind}$-set.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133962989","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On BMRN*-colouring of planar digraphs","authors":"Julien Bensmail, Foivos Fioravantes","doi":"10.46298/dmtcs.5798","DOIUrl":"https://doi.org/10.46298/dmtcs.5798","url":null,"abstract":"International audience\u0000 \u0000 In a recent work, Bensmail, Blanc, Cohen, Havet and Rocha, motivated by applications for TDMA scheduling problems, have introduced the notion of BMRN*-colouring of digraphs, which is a type of arc-colouring with particular colouring constraints. In particular, they gave a special focus to planar digraphs. They notably proved that every planar digraph can be 8-BMRN*-coloured, while there exist planar digraphs for which 7 colours are needed in a BMRN*-colouring. They also proved that the problem of deciding whether a planar digraph can be 3-BMRN*-coloured is NP-hard. In this work, we pursue these investigations on planar digraphs, in particular by answering some of the questions left open by the authors in that seminal work. We exhibit planar digraphs needing 8 colours to be BMRN*-coloured, thus showing that the upper bound of Bensmail, Blanc, Cohen, Havet and Rocha cannot be decreased in general. We also generalize their complexity result by showing that the problem of deciding whether a planar digraph can be k-BMRN*-coloured is NP-hard for every k ∈ {3,...,6}. Finally, we investigate the connection between the girth of a planar digraphs and the least number of colours in its BMRN*-colourings.\u0000","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133878207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Automatic sequences: from rational bases to trees","authors":"M. Rigo, Manon Stipulanti","doi":"10.46298/dmtcs.8455","DOIUrl":"https://doi.org/10.46298/dmtcs.8455","url":null,"abstract":"The $n$th term of an automatic sequence is the output of a deterministic\u0000finite automaton fed with the representation of $n$ in a suitable numeration\u0000system. In this paper, instead of considering automatic sequences built on a\u0000numeration system with a regular numeration language, we consider those built\u0000on languages associated with trees having periodic labeled signatures and, in\u0000particular, rational base numeration systems. We obtain two main\u0000characterizations of these sequences. The first one is concerned with $r$-block\u0000substitutions where $r$ morphisms are applied periodically. In particular, we\u0000provide examples of such sequences that are not morphic. The second\u0000characterization involves the factors, or subtrees of finite height, of the\u0000tree associated with the numeration system and decorated by the terms of the\u0000sequence.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132442910","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Positional Marked Patterns in Permutations","authors":"S. Thamrongpairoj, J. Remmel","doi":"10.46298/dmtcs.7171","DOIUrl":"https://doi.org/10.46298/dmtcs.7171","url":null,"abstract":"We define and study positional marked patterns, permutations $tau$ where one\u0000of elements in $tau$ is underlined. Given a permutation $sigma$, we say that\u0000$sigma$ has a $tau$-match at position $i$ if $tau$ occurs in $sigma$ in\u0000such a way that $sigma_i$ plays the role of the underlined element in the\u0000occurrence. We let $pmp_tau(sigma)$ denote the number of positions $i$ which\u0000$sigma$ has a $tau$-match. This defines a new class of statistics on\u0000permutations, where we study such statistics and prove a number of results. In\u0000particular, we prove that two positional marked patterns $1underline{2}3$ and\u0000$1underline{3}2$ give rise to two statistics that have the same distribution.\u0000The equidistibution phenomenon also occurs in other several collections of\u0000patterns like $left {1underline{2}3 , 1underline{3}2 right }$, and $left\u0000{ 1underline234, 1underline243, underline2134, underline2 1 4 3 right\u0000}$, as well as two positional marked patterns of any length $n$: $left {\u00001underline 2tau , underline 21tau right }$.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"128 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124539295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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