In 1966, Gallai asked whether all longest paths in a connected graph have nonempty intersection. The answer to this question is not true general and various counterexamples been found. However, there positive solution Gallai’s for many well-known classes of graphs such as split graphs, series–parallel 2K2-free graphs. Split were proven be Hamiltonian under given toughness conditions. This obser...

Let G be a connected graph on n vertices. The Gallai number Gal(G) of is the size smallest set vertices that meets every maximum path in G. Grünbaum constructed with Gal(G)=3. Very recently, Long, Milans, and Munaro, proved Gal(G)≤8n3/4. This was first sub-linear upper bound terms n. We improve their to Gal(G)≤5n2/3. also tighten more general result Long et al. For multigraph M, we prove if L(M...

A set of matrices over the integers is said to be length k mortal with k positive integer if the zero matrix can be expressed as a product of length k of matrices in the set The set is said to be mortal if it is length k mortal for some nite k We show that the problem of deciding whether a pair of integer matrices is mortal is undecidable and that the problem of deciding for a given k whether a...

A path decomposition of a graph G is a collection of edge-disjoint paths of G that covers the edge set of G. Gallai (1968) conjectured that every connected graph on n vertices admits a path decomposition of cardinality at most ⌊(n + 1)/2⌋. Gallai’s Conjecture has been verified for many classes of graphs. In particular, Lovász (1968) verified this conjecture for graphs with at most one vertex wi...

Predictions of localized Majorana modes, and ideas for manipulating these degrees freedom, are the two key ingredients in proposals physical platforms quantum computation. Several envisage a scalable network such modes coupled bilinearly to each other by quantum-mechanical mixing amplitudes. Here, we develop theoretical framework characterizing collective topologically protected zero-energy fer...

Let the matching polynomial of a graph G be denoted by μ(G, x). A graph G is said to be θ-super positive if μ(G, θ) 6= 0 and μ(G \ v, θ) = 0 for all v ∈ V (G). In particular, G is 0-super positive if and only if G has a perfect matching. While much is known about 0-super positive graphs, almost nothing is known about θsuper positive graphs for θ 6= 0. This motivates us to investigate the struct...

Generalizing Kaneko’s long path packing problem, Hartvigsen, Hell and Szabó consider a new type of undirected graph packing problem, called the k-piece packing problem. A k-piece is a simple, connected graph with highest degree exactly k so in the case k = 1 we get the classical matching problem. They give a polynomial algorithm, a Tutte-type characterization and a Berge-type minimax formula fo...

Large graphs are sometimes studied through their degree sequences (power law or regular graphs). We study graphs that are uniformly chosen with a given degree sequence. Under mild conditions, it is shown that sequences of such graphs have graph limits in the sense of Lovász and Szegedy with identifiable limits. This allows simple determination of other features such as the number of triangles. ...