We present an algorithm to find a minimum cover (partition) of a hypergraph using hyperforests. In a more general setting, given a demand vector associated with the hyperedges, we seek a minimum cover (partition) of this demand. This approach is used to express a vector in the hypergraphic matroid polytope as a convex combination of incidence vectors of hyperforests.
An (κ,τ)-regular set in a graph Γ is a nonempty proper subset D of the vertex set V(Γ) such that every vertex in D has exactly κ neighbors in D and every vertex in V(Γ)∖D has exactly τ neighbors in D, where κ and τ are nonnegative integers. Our main contributions are threefold. First, we establish a complete characterization of the existence of all possible (κ,τ)-regular sets in generalized Petersen graphs, providing a necessary and sufficient condition for each admissible pair (κ,τ). Second, we give a complete classification of all (1,2)- and (1,3)-regular sets in generalized Petersen graphs. Finally, as an application, we resolve the existence problem of perfect 2-colorings of generalized Petersen graphs.
Given an undirected connected graph G=(V,E) with an edge weight function w:E→R+, and k positive reals w1,w2,…,wk with ∑i=1kwi=∑v∈Ew(e) and mini∈[k]wi≥maxe∈Ew(e), Connected k-Edge Partition (CEP) asks to find a partition {E1,E2,…,Ek} of E such that (i) Ei induces a connected graph in G, and (ii) ∑e∈Eiw(e)=wi for all i∈[k]. Even in the unweighted case, i.e., the case where w(e)=1 for all e∈E and wi∈N for all i∈[k], it is known to be NP-complete whether CEP has a feasible partition for any fixed k≥2 Dyer and Frieze, (1985). In this paper, we show that if wi≥2maxe∈Ew(e) for all i∈[k], then we can find in O(k2|E|) time a “2-approximate” partition for CEP, i.e., a partition of E satisfying (i) and (ii’) 12wi≤∑e∈Eiw(e)≤2wi for all i∈[k], while there exists a graph G which has no better than k-approximate partition in general. Also, we show that the unweighted case always has a 2-approximate partition even for general graphs, and such a partition can be found in O(k2|E|) time.
We construct a bijection between (i) the set of subgraphs of C2n (the cyclic graph with 2n vertices) that have n edges and k connected components and (ii) the set of paths from (0,0) to (n,n) composed of any of the steps (1,0), (1,1), and (0,1) that do not start with (1,1) and contain exactly k-(1,0) steps. This work is motivated by the work of Selig (2023) who proved that the number of recurrent states of the stochastic sandpile model on C2n is equal to the cardinalities of these sets described above.
We study the Turán numbers of 3-graphs avoiding 3-graphs F and Ms+13, a matching of size s+1. Denote by K43− the 3-graph on vertex set [4] with edges 123,124,234. The asymptotic value of ex(n,{K43−,Ms+13}) for s≥3 and sufficiently large n was determined by Chen et al. (2025). In this paper we establish the exact value of ex(n,{K43−,Ms+13}) in the case s=3 for all n≥400.
The resonance graph of a hexagonal system is connected, which shows that a perfect matching can be transformed into any other perfect matchings by a series of flips along hexagons. However, the resonance graph of a coronoid system (with holes) is not necessarily connected. Saldanha et al. (Discrete Comput. Geom. 14 (1995) 207-233) used homology and cohomology theory to obtain three versions of criteria for two tilings of a quadriculated region in the plane to be in the same connected component of the flip graph. Inspiblack by the combinatorial version, in this paper we use a purely graph-theoretical approach to give a criterion in terms of simple invariant—flow across cuts between holes/exterior face for two perfect matchings of a coronoid system G to be in the same connected component of its resonance graph. As a corollary we obtain a criterion for the resonance graph of a coronoid system to be connected. We also discuss whether such criteria are applicable to nanotubes, and construct a nanotube whose resonance graph is connected, which disproves a conjecture proposed by Tratnik et al. (MATCH Commun. Math. Comput. Chem. 74 (2015) 175-186).
Let G be a connected graph and S⊆V(G) with |S|≥2. A tree T in G is called an S-tree if S⊆V(T). Two S-trees T1 and T2 are called internally disjoint if E(T1)∩E(T2)=0̸ and V(T1)∩V(T2)=S. For an integer k with 2≤k≤n, the generalized k-connectivity of a graph G is defined as κk(G)=min{κG(S)|S⊆V(G) and |S|=k}, where κG(S) denotes the maximum number of internally disjoint S-trees in G. The generalized k-connectivity extends traditional connectivity and serves as a crucial measure for evaluating the reliability and fault tolerance of connecting any k vertices in G. In this paper, we mainly investigate the generalized 4-connectivity of a family of regular graph Gn, which improves the known results about generalized 3-connectivity of Gn in Zhao et al., (2021). For a reason that the alternating group network ANn, the star graph Sn and pancake graphs Pn are special cases of the regular graph Gn, as applications of the main result, we obtain that κ4(ANn)=n−2 for n≥4, κ4(Sn)=n−2 for n≥3 and κ4(Pn)=n−2 for n≥3.
A graph G=(V,E) is called (p,q)-bipartite, if there exists a partition of the vertex set V into two sets V1 and V2 such that each vertex u∈V1 has at least p neighbors in V2 and each vertex v∈V2 has at least q neighbors in V1. The corresponding decision problem (p,q)-BIPARTITION asks whether a given graph G is (p,q)-bipartite. Bang-Jensen et al. (JGT, 2019) showed that (p,q)-BIPARTITION is NP-complete if and only if p+q≥4. In this paper, we provide a complexity dichotomy for (p,q)-BIPARTITION for the class of planar graphs, except for the case p=q=2. We present a similar result for the class of 1-planar graphs, except for eight cases. We further prove that, for k≥1, (p,q)-BIPARTITION is NP-complete for the class of k-planar graphs if and only if p+q≥4 and there exists at least one (p,q)-bipartite k-planar graph, as a generalization of the above results. We also prove that a k-planar (p,q)-bipartite graph exists if and only if a k-planar (p,q)-biregular graph exists.
