Mathematics

A bridgeless cubic graph G is said to have a 2-bisection if there exists a 2-vertex-colouring of G (not necessarily proper) such that: (i) the colour classes have the same cardinality, and (ii) the monochromatic components are either an isolated vertex or an edge. In 2016, Ban and Linial conjectured that every bridgeless cubic graph, apart from the well-known Petersen graph, admits a 2-bisection. In the same paper it was shown that every Class I bridgeless cubic graph admits such a bisection. The Class II bridgeless cubic graphs which are critical to many conjectures in graph theory are snarks, in particular, those with excessive index at least 5, that is, whose edge-set cannot be covered by four perfect matchings. Moreover, Esperet et al. state that a possible counterexample to Ban--Linial's Conjecture must have circular flow number at least 5. The same authors also state that although empirical evidence shows that several graphs obtained from the Petersen graph admit a 2-bisection, they can offer nothing in the direction of a general proof. Despite some sporadic computational results, until now, no general result about snarks having excessive index and circular flow number both at least 5 has been proven. In this work we show that treelike snarks, which are an infinite family of snarks heavily depending on the Petersen graph and with both their circular flow number and excessive index at least 5, admit a 2-bisection.

The original Beck conjecture, now a theorem due to Andrews, states that the difference in the number of parts in all partitions into odd parts and the number of parts in all strict partitions is equal to the number of partitions whose set of even parts has one element, and also to the number of partitions with exactly one part repeated. This is a companion identity to Euler's identity. The theorem has been generalized by Yang to a companion identity to Glaisher's identity. Franklin generalized Glaisher's identity, and in this article, we provide a Beck-type companion identity to Franklin's identity and prove it both analytically and combinatorially. Andrews' and Yang's respective theorems fit naturally into this very general frame. We also discuss a generalization to Franklin's identity of the second Beck-type companion identity proved by Andrews and Yang in their respective work.

Characterizing the graph classes such that, on n -vertex m -edge graphs in the class, we can compute the diameter faster than in O(nm) time is an important research problem both in theory and in practice. We here make a new step in this direction, for some metrically defined graph classes. Specifically, a subgraph H of a graph G is called a retract of G if it is the image of some idempotent endomorphism of G . Two necessary conditions for H being a retract of G is to have H is an isometric and isochromatic subgraph of G . We say that H is an absolute retract of some graph class C if it is a retract of any G?�C of which it is an isochromatic and isometric subgraph. In this paper, we study the complexity of computing the diameter within the absolute retracts of various hereditary graph classes. First, we show how to compute the diameter within absolute retracts of bipartite graphs in randomized O ~ (m n ??????) time. For the special case of chordal bipartite graphs, it can be improved to linear time, and the algorithm even computes all the eccentricities. Then, we generalize these results to the absolute retracts of k -chromatic graphs, for every fixed k?? . Finally, we study the diameter problem within the absolute retracts of planar graphs and split graphs, respectively.

Let P,Q be longest paths in a simple graph. We analyze the possible connections between the components of P?�Q??V(P)?�V(Q)) and introduce the notion of a bi-traceable graph. We use the results for all the possible configurations of the intersection points when #V(P)?�V(Q)?? in order to prove that if the intersection of three longest paths P,Q,R is empty, then #(V(P)?�V(Q))?? . We also prove Hippchen's conjecture for k?? : If a graph G is k -connected for k?? , and P and Q are longest paths in G , then #(V(P)?�V(Q))?? .

We provide a canonical decomposition for a class of bipartite grafts known as combs. As every bipartite graft is a recursive combination of combs, our results provides a canonical decomposition for general bipartite grafts. Our new decomposition is by definition a generalization of the classical canonical decomposition in matching theory, that is, the Dulmage-Mendelsohn decomposition for bipartite graphs with perfect matchings. However, it exhibits much more complicated structure than its classical counterpart. It is revealed from our results that bipartite grafts has a canonical structure that is analogous to the cathedral decomposition for nonbipartite graphs with perfect matchings.

Permutation statistics $\wnm$ and $\rlm$ are both arising from permutation tableaux. $\wnm$ was introduced by Chen and Zhou, which was proved equally distributed with the number of unrestricted rows of a permutation tableau. While $\rlm$ is showed by Nadeau equally distributed with the number of 1 's in the first row of a permutation tableau. In this paper, we investigate the joint distribution of $\wnm$ and $\rlm$. Statistic $(\rlm,\wnm,\rlmin,\des,(\underline{321}))$ is shown equally distributed with $(\rlm,\rlmin,\wnm,\des,(\underline{321}))$ on S n . Then the generating function of $(\rlm,\wnm)$ follows. An involution is constructed to explain the symmetric property of the generating function. Also, we study the triple statistic $(\wnm,\rlm,\asc)$, which is shown to be equally distributed with $(\rlmax-1,\rlmin,\asc)$ as studied by Josuat-Verg e ` s. The main method we adopt throughout the paper is constructing bijections based on a block decomposition of permutations.

The b-chromatic number b(G) of a graph G is the maximum k for which G has a proper vertex coloring using k colors such that each color class contains at least one vertex adjacent to a vertex of every other color class. In this paper, we mainly investigate on one of the open problems given in [P. Francis, S. Francis Raj, On b-coloring of powers of hypercubes, Discrete Appl. Math. 225 (2017) 74-86.]. As a consequence, we have obtained an upper bound for the b-chromatic number of some powers of hypercubes. This turns out to be an improvement of the already existing bound in [P. Francis, S. Francis Raj, On b-coloring of powers of hypercubes, Discrete Appl. Math. 225 (2017) 74-86.]. Further, we have determined a lower bound for the b-chromatic number of some powers of the Hamming graph, a generalization of the hypercube.

A complex unit gain graph ( T -gain graph), Φ=(G,?) is a graph where the function ? assigns a unit complex number to each orientation of an edge of G , and its inverse is assigned to the opposite orientation. A T -gain graph Φ is balanced if the product of the edge gains of each cycle (with a fixed orientation) is 1 . Signed graphs are special cases of T -gain graphs. The adjacency matrix of Φ , denoted by A(Φ) is defined canonically. The gain Laplacian for Φ is defined as L(Φ)=D(Φ)?�A(Φ) , where D(Φ) is the diagonal matrix with diagonal entries are the degrees of the vertices of G . The minimum number of vertices (resp., edges) to be deleted from Φ in order to get a balanced gain graph the frustration number (resp, frustration index). We show that frustration number and frustration index are bounded below by the smallest eigenvalue of L(Φ) . We provide several lower and upper bounds for extremal eigenvalues of L(Φ) in terms of different graph parameters such as the number of edges, vertex degrees, and average 2 -degrees. The signed graphs are particular cases of the T -gain graphs, all the bounds established in paper hold for signed graphs. Most of the bounds established here are new for signed graphs. Finally, we perform comparative analysis for all the obtained bounds in the paper with the state-of-the-art bounds available in the literature for randomly generated Erd?s-Reýni graphs. Some of the major highlights of our paper are the gain-dependent bounds, limit convergence of the bounds to the extremal eigenvalues, and optimal extremal bounds obtained by posing optimization problems to achieve the best possible bounds.

A d -dimensional brick is a set I 1 ??��?I d where each I i is an interval. Given a brick B , a brick partition of B is a partition of B into bricks. A brick partition P d of a d -dimensional brick is k -piercing if every axis-parallel line intersects at least k bricks in P d . Bucic et al. explicitly asked the minimum size p(d,k) of a k -piercing brick partition of a d -dimensional brick. The answer is known to be 4(k??) when d=2 . Our first result almost determines p(3,k) . Namely, we construct a k -piercing brick partition of a 3 -dimensional brick with 12k??5 parts, which is off by only 1 from the known lower bound. As a generalization of the above question, we also seek the minimum size s(d,k) of a brick partition P d of a d -dimensional brick where each axis-parallel plane intersects at least k bricks in P d . We resolve the question in the 3 -dimensional case by determining s(3,k) for all k .

We determine the maximum number of induced copies of a 5-cycle in a graph on n vertices for every n . Every extremal construction is a balanced iterated blow-up of the 5-cycle with the possible exception of the smallest level where for n=8 , the Möbius ladder achieves the same number of induced 5-cycles as the blow-up of a 5-cycle on 8 vertices. This result completes work of Balogh, Hu, Lidický, and Pfender [Eur. J. Comb. 52 (2016)] who proved an asymptotic version of the result. Similarly to their result, we also use the flag algebra method but we extend its use to small graphs.