Journal Of Combinatorial Theory Research Articles

Blazing a Trail via Matrix Multiplications: A Faster Algorithm for Non-shortest Induced Paths

For vertices u and v of an n-vertex graph G, a uv-trail of G is an induced uv-path of G that is not a shortest uv-path of G. Berger, Seymour, and Spirkl [Discrete Mathematics 2021] gave the previously only known polynomial-time algorithm, running in O(n18) time, to either output a uv-trail of G or ensure that G admits no uv-trail. We reduce the complexity to the time required to perform a poly-logarithmic number of multiplications of n2×n2 Boolean matrices, leading to a largely improved O(n4.75)-time algorithm. Our result improves the previous O(n21)-time algorithm by Cook, Horsfield, Preissmann, Robin, Seymour, Sintiari, Trotignon, and Vušković [Journal of Combinatorial Theory, Series B, 2024] for recognizing graphs with all holes the same length, and reduces the running time to O(n7.75).

KOLMOGOROV BOUNDS FOR THE NORMAL APPROXIMATION OF THE NUMBER OF TRIANGLES IN THE ERDŐS–RÉNYI RANDOM GRAPH

We bound the error for the normal approximation of the number of triangles in the Erdős–Rényi random graph with respect to the Kolmogorov metric. Our bounds match the best available Wasserstein bounds obtained by Barbouret al.[(1989). A central limit theorem for decomposable random variables with applications to random graphs.Journal of Combinatorial Theory, Series B47: 125–145], resolving a long-standing open problem. The proofs are based on a new variant of the Stein–Tikhomirov method—a combination of Stein's method and characteristic functions introduced by Tikhomirov [(1976). The rate of convergence in the central limit theorem for weakly dependent variables.Vestnik Leningradskogo Universiteta158–159, 166].

On Permutation Weights and q-Eulerian Polynomials

Weights of permutations were originally introduced by Dugan, Glennon, Gunnells, and Steingr\'imsson (Journal of Combinatorial Theory, Series A 164:24-49, 2019) in their study of the combinatorics of tiered trees. Given a permutation $\sigma$ viewed as a sequence of integers, computing the weight of $\sigma$ involves recursively counting descents of certain subpermutations of $\sigma$. Using this weight function, one can define a $q$-analog $E_n(x,q)$ of the Eulerian polynomials. We prove two main results regarding weights of permutations and the polynomials $E_n(x,q)$. First, we show that the coefficients of $E_n(x, q)$ stabilize as $n$ goes to infinity, which was conjectured by Dugan, Glennon, Gunnells, and Steingr\'imsson (Journal of Combinatorial Theory, Series A 164:24-49, 2019), and enables the definition of the formal power series $W_d(t)$, which has interesting combinatorial properties. Second, we derive a recurrence relation for $E_n(x, q)$, similar to the known recurrence for the classical Eulerian polynomials $A_n(x)$. Finally, we give a recursive formula for the numbers of certain integer partitions and, from this, conjecture a recursive formula for the stabilized coefficients mentioned above.

Hypergraph coloring up to condensation

Improving a result of Dyer, Frieze and Greenhill [Journal of Combinatorial Theory, Series B, 2015], we determine the q‐colorability threshold in random k‐uniform hypergraphs up to an additive error of , where . The new lower bound on the threshold matches the “condensation phase transition” predicted by statistical physics considerations [Krzakala et al., PNAS 2007].

A note on the maximum number of triangles in a C5‐free graph

AbstractWe prove that the maximum number of triangles in a ‐free graph on vertices is at most , improving an estimate of Alon and Shikhelman.

A note on the maximum number of triangles in a C5-free graph

We prove that the maximum number of triangles in a C5-free graph on n vertices is at most 122(1+o(1))n3/2, improving an estimate of Alon and Shikhelman [Alon, N. and C. Shikhelman, Many T copies in H-free graphs. Journal of Combinatorial Theory, Series B 121 (2016) 146-172].

How does the core sit inside the mantle?

The k-core, defined as the largest subgraph of minimum degree k, of the random graph G(n,p) has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [Journal of Combinatorial Theory, Series B 67 (1996) 111–151] determined the threshold dk for the appearance of an extensive k-core. Here we derive a multi-type Galton-Watson branching process that describes precisely how the k-core is “embedded” into the random graph for any k≥3 and any fixed average degree d=np>dk. This generalises prior results on, e.g., the internal structure of the k-core.

Induced minors and well-quasi-ordering

A graph H is an induced minor of a graph G if it can be obtained from an induced subgraph of G by contracting edges. Otherwise, G is said to be H-induced minor-free. Robin Thomas showed in [Graphs without K4and well-quasi-ordering, Journal of Combinatorial Theory, Series B, 38(3):240 – 247, 1985] that K4-induced minor-free graphs are well-quasi ordered by induced minors.We provide a dichotomy theorem for H-induced minor-free graphs and show that the class of H-induced minor-free graphs is well-quasi-ordered by the induced minor relation if and only if H is an induced minor of the gem (the path on 4 vertices plus a dominating vertex) or of the graph obtained by adding a vertex of degree 2 to the complete graph on 4 vertices.Similar dichotomy results were previously given by Guoli Ding in [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489–502, 1992] for subgraphs and Peter Damaschke in [Induced subgraphs and well-quasi-ordering, Journal of Graph Theory, 14(4):427–435, 1990] for induced subgraphs.

A dual version of the Brooks group coloring theorem

Let G be a 2-edge-connected undirected graph, A be an (additive) Abelian group, and A∗=A−{0}. A graph G is A-connected if G has an orientation D(G) such that for every mapping b:V(G)↦A satisfying ∑v∈V(G)b(v)=0, there is a function f:E(G)↦A∗ such that for each vertex v∈V(G), the sum of f over the edges directed out from v minus the sum of f over the edges directed into v equals b(v). For a 2-edge-connected graph G, define Λg(G)=min{k: for any Abelian group A with |A|≥k, G is A-connected }. Let P denote a path in G, let βG(P) be the minimum length of a circuit containing P, and let βi(G) be the maximum of βG(P) over paths of length i in G. We show that Λg(G)≤βi(G)+1 for any integer i>0 and for any 2-connected graph G. Partial solutions toward determining the graphs for which equality holds were obtained by Fan et al. in [G. Fan, H.-J. Lai, R. Xu, C.-Q. Zhang, C. Zhou, Nowhere-zero 3-flows in triangularly connected graphs, Journal of Combinatorial Theory, Series B 98 (6) (2008) 1325–1336], among others. In this paper, we completely determine all graphs G with Λg(G)=β2(G)+1.

Minimum Weight $H$-Decompositions of Graphs: The Bipartite Case

Given graphs $G$ and $H$ and a positive number $b$, a weighted $(H,b)$-decomposition of $G$ is a partition of the edge set of $G$ such that each part is either a single edge or forms an $H$-subgraph. We assign a weight of $b$ to each $H$-subgraph in the decomposition and a weight of 1 to single edges. The total weight of the decomposition is the sum of the weights of all elements in the decomposition. Let $\phi(n,H,b)$ be the the smallest number such that any graph $G$ of order $n$ admits an $(H,b)$-decomposition with weight at most $\phi(n,H,b)$. The value of the function $\phi(n,H,b)$ when $b=1$ was determined, for large $n$, by Pikhurko and Sousa [Minimum $H$-Decompositions of Graphs, Journal of Combinatorial Theory, B, 97 (2007), 1041–1055.] Here we determine the asymptotic value of $\phi(n,H,b)$ for any fixed bipartite graph $H$ and any value of $b$ as $n$ tends to infinity.

Recognizing Helly Edge-Path-Tree graphs and their clique graphs

We present a unifying procedure for recognizing intersection graphs of Helly families of paths in a tree and their clique graphs. The Helly property makes it possible to look at these recognition problems as variants of the Graph Realization Problem, namely, the problem of recognizing Edge-Path-Tree matrices. Our result heavily relies on the notion of pie introduced in [M.C. Golumbic, R.E. Jamison, The edge intersection graphs of paths in a tree, Journal of Combinatorial Theory, Series B 38 (1985) 8–22] and on the observation that Helly Edge-Path-Tree matrices form a self-dual class of Helly matrices. Coupled to the notion of reduction presented in the paper, these facts are also exploited to reprove and slightly refine some known results for Edge-Path-Tree graphs.

Yueh-Gin Gung and Dr. Charles Y. Hu Award for 2011 to Joseph A. Gallian for Distinguished Service to Mathematics

The two themes that run through Joseph A. Gallian’s service to mathematics are (a) encouraging young mathematicians and helping them to develop successful careers and (b) communicating mathematics to the widest possible audience. He was one of the early proponents of undergraduates conducting mathematical research, and his REU at Duluth, which began in 1977, is widely regarded as the premier REU. The quality of the work at Joe’s REU is evidenced by the 160 papers by participants that grew out of their REU work. These papers have appeared in such journals as Crelle’s Journal, Journal of Algebra, Journal of Combinatorial Theory, Discrete Mathematics, Applied Discrete Mathematics, Annals of Discrete Mathematics, and Journal of Graph Theory. The REU, along with Joe’s continuing contact with its participants, makes an important contribution to developing the next generations of mathematicians. Participants have included some prominent mathematicians, whose careers this REU helped to shape. Joe is also an inspiration to a generation of mathematicians who involve students in high-quality undergraduate research in mathematics. Not only is Joe successful with his own REU, but he is generous with his time and advice to help others to set up REUs. In 2002, Joe was recognized by the Council on Undergraduate Research with their Fellow Award, given to members who have demonstrated sustained excellence in research with undergraduates.

Distinct squares in run-length encoded strings

Squares are strings of the form w w where w is any nonempty string. Two squares w w and w ′ w ′ are of different types if and only if w ≠ w ′ . Fraenkel and Simpson [Avieri S. Fraenkel, Jamie Simpson, How many squares can a string contain? Journal of Combinatorial Theory, Series A 82 (1998) 112–120] proved that the number of square types contained in a string of length n is bounded by O ( n ) . The set of all different square types contained in a string is called the vocabulary of the string. If a square can be obtained by a series of successive right-rotations from another square, then we say the latter covers the former. A square is called a c-square if no square with a smaller index can cover it and it is not a trivial square. The set containing all c -squares is called the covering set. Note that every string has a unique covering set. Furthermore, the vocabulary of the covering set are called c - vocabulary. In this paper, we prove that the cardinality of c -vocabulary in a string is less than 14 3 N , where N is the number of runs in this string.

On the set covering polyhedron of circulant matrices

A well known family of minimally nonideal matrices is the family of the incidence matrices of chordless odd cycles. A natural generalization of these matrices is given by the family of circulant matrices. Ideal and minimally nonideal circulant matrices have been completely identified by Cornuéjols and Novick [G. Cornuéjols, B. Novick, Ideal 0 - 1 matrices, Journal of Combinatorial Theory B 60 (1994) 145–157]. In this work we classify circulant matrices and their blockers in terms of the inequalities involved in their set covering polyhedra. We exploit the results due to Cornuéjols and Novick in the above-cited reference for describing the set covering polyhedron of blockers of circulant matrices. Finally, we point out that the results found on circulant matrices and their blockers present a remarkable analogy with a similar analysis of webs and antiwebs due to Pêcher and Wagler [A. Pêcher, A. Wagler, A construction for non-rank facets of stable set polytopes of webs, European Journal of Combinatorics 27 (2006) 1172–1185; A. Pêcher, A. Wagler, Almost all webs are not rank-perfect, Mathematical Programming Series B 105 (2006) 311–328] and Wagler [A. Wagler, Relaxing perfectness: Which graphs are ‘Almost’ perfect?, in: M. Groetschel (Ed.), The Sharpest Cut, Impact of Manfred Padberg and his work, in: SIAM/MPS Series on Optimization, vol. 4, Philadelphia, 2004; A. Wagler, Antiwebs are rank-perfect, 4OR 2 (2004) 149–152].

Integrality properties of edge path tree families

An Edge Path Tree (EPT) family is a family whose members are edge sets of paths in a tree. Relying on the notion of Pie introduced in [M.C. Golumbic, R.E. Jamison, The edge intersection graphs of paths in a tree, Journal of Combinatorial Theory, Series B 38 (1985) 8–22], we characterize Ideal and Mengerian EPT families. In particular, we show that an EPT family is Ideal if and only if it is Mengerian. If, in addition, the EPT family is uniform, then it is Ideal if and only if it is Unimodular. The latter equivalence generalizes the well-known fact that the edge set of a graph is an Ideal clutter if and only if the graph is bipartite.

A combinatorial approach to jumping particles: The parallel TASEP

AbstractIn this article, we continue the combinatorial study of models of particles jumping on a row of cells which we initiated with the standard totally asymmetric simple exclusion process or TASEP (Duchi and Schaeffer, Journal of Combinatorial Theory, Series A, 110(2005), 1–29). We consider here the parallel TASEP, in which particles can jump simultaneously. On the one hand, the interest in this process comes from highway traffic modeling: it is the only solvable special case of the Nagel‐Schreckenberg automaton, the most popular model in that context. On the other hand, the parallel TASEP is of some theoretical interest because the derivation of its stationary distribution, as appearing in the physics literature, is harder than that of the standard TASEP.We offer here an elementary derivation that extends the combinatorial approach we developed for the standard TASEP. In particular, we show that this stationary distribution can be expressed in terms of refinements of Catalan numbers. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008

Disjoint paths in symmetric digraphs

Given a number of requests ℓ , we propose a polynomial-time algorithm for finding ℓ disjoint paths in a symmetric directed graph. It is known that the problem of finding ℓ ≥ 2 disjoint paths in a directed graph is NP-hard [S. Fortune, J. Hopcroft, J. Wyllie, The directed subgraph homeomorphism problem, Journal of Theoretical Computer Science 10 (2) (1980) 111–121]. However, by studying minimal solutions it turns out that only a finite number of configurations are possible in a symmetric digraph. We use Robertson and Seymour’s polynomial-time algorithm [N. Robertson, P. D. Seymour, Graph minors xiii. The disjoint paths problem, Journal of Combinatorial Theory B (63) (1995) 65–110] to check the feasibility of each configuration.

On Matroids and Nonideal Secret Sharing

Secret-sharing schemes are a tool used in many cryptographic protocols. In these schemes, a dealer holding a secret string distributes shares to the parties such that only authorized subsets of participants can reconstruct the secret from their shares. The collection of authorized sets is called an access structure. An access structure is ideal if there is a secret-sharing scheme realizing it such that the shares are taken from the same domain as the secrets. Brickell and Davenport (Journal of Cryptology, 1991) have shown that ideal access structures are closely related to matroids. They give a necessary condition for an access structure to be ideal-the access structure must be induced by a matroid. Seymour (Journal of Combinatorial Theory B, 1992) has proved that the necessary condition is not sufficient: There exists an access structure induced by a matroid that does not have an ideal scheme. The research on access structures induced by matroids is continued in this work. The main result in this paper is strengthening the result of Seymour. It is shown that in any secret-sharing scheme realizing the access structure induced by the Vamos matroid with domain of the secrets of size k, the size of the domain of the shares is at least k + Omega(radic(k)). The second result considers nonideal secret-sharing schemes realizing access structures induced by matroids. It is proved that the fact that an access structure is induced by a matroid implies lower and upper bounds on the size of the domain of shares of subsets of participants even in nonideal schemes (as long as the shares are still relatively short). This generalized results of Brickell and Davenport for ideal schemes. Finally, an example of a nonideal access structure that is nearly ideal is presented.

A 3-color Theorem on Plane Graphs without 5-circuits

In this paper, we prove that every plane graph without 5-circuits and without triangles of distance less than 3 is 3-colorable. This improves the main result of Borodin and Raspaud [Borodin, O. V., Raspaud, A.: A sufficient condition for planar graphs to be 3-colorable. Journal of Combinatorial Theory, Ser. B, 88, 17–27 (2003)], and provides a new upper bound to their conjecture.

The Faà di Bruno formula and determinant identities

The Faà di Bruno formula on higher derivatives of composite functions and formal power series iteration is applied to the combinatorial determinant evaluation. Several determinant identities on composite function derivations and formal power series iterations, as well as their weighted versions, are established. They contain, as particular cases, the determinant evaluations due to Mina (Mina, L., 1905, Formule generali delle successive d'una funzione espresse mediante quelle della sua inversa. Giornale di Matematiche, 43, 196–212.), Kedlaya (Kedlaya, K.S., 2000, Another combinatorial determinant. Journal of Combinatorial Theory (A), 90, 221–223.), and Wilf (Wilf, H.S., A combinatorial determinant. http://www.cis.upenn.edu/~wilf.).