Some Properties of Yao Y\u3csub\u3e4\u3c/sub\u3e Subgraphs

The Yao graph for k = 4, Y4, is naturally partitioned into four subgraphs, one per quadrant. We show that the subgraphs for one quadrant differ from the subgraphs for two adjacent quadrants in three properties: planarity, connectedness, and whether the directed graphs are spanners

Graph Coloring Problems and Group Connectivity

1. Group connectivity. Let A be an abelian group and let iA(G) be the smallest positive integer m such that Lm(G) is A-connected. A path P of G is a normal divalent path if all internal vertices of P are of degree 2 in G and if |E(P)|= 2, then P is not in a 3-cycle of G. Let l(G) = max{lcub}m : G has a normal divalent path of length m{rcub}. We obtain the following result. (i) If |A| ≥ 4, then iA( G) ≤ l(G). (ii) If | A| ≥ 4, then iA(G) ≤ |V(G)| -- Delta(G). (iii) Suppose that |A| ≥ 4 and d = diam( G). If d ≤ |A| -- 1, then iA(G) ≤ d; and if d ≥ |A|, then iA(G) ≤ 2d -- |A| + 1. (iv) iZ 3 (G) ≤ l(G) + 2. All those bounds are best possible.;2. Modulo orientation. A mod (2p + 1)-orientation D is an orientation of G such that d +D(v) = d--D(v) (mod 2p + 1) for any vertex v ∈ V ( G). We prove that for any integer t ≥ 2, there exists a finite family F = F(p, t) of graphs that do not have a mod (2p + 1)-orientation, such that every graph G with independence number at most t either admits a mod (2p+1)-orientation or is contractible to a member in F. In particular, the graph family F(p, 2) is determined, and our results imply that every 8-edge-connected graph G with independence number at most two admits a mod 5-orientation.;3. Neighbor sum distinguishing total coloring. A proper total k-coloring &phis; of a graph G is a mapping from V(G) ∪ E(G) to {lcub}1,2, . . .,k{rcub} such that no adjacent or incident elements in V(G) ∪ E( G) receive the same color. Let m&phis;( v) denote the sum of the colors on the edges incident with the vertex v and the color on v. A proper total k-coloring of G is called neighbor sum distinguishing if m &phis;(u) ≠ m&phis;( v) for each edge uv ∈ E( G ). Let chitSigma(G) be the neighbor sum distinguishing total chromatic number of a graph G. Pilsniak and Wozniak conjectured that for any graph G, chitSigma( G) ≤ Delta(G) + 3. We show that if G is a graph with treewidth ℓ ≥ 3 and Delta(G) ≥ 2ℓ + 3, then chitSigma( G) + ℓ -- 1. This upper bound confirms the conjecture for graphs with treewidth 3 and 4. Furthermore, when ℓ = 3 and Delta ≥ 9, we show that Delta(G)+1 ≤ chit Sigma(G) ≤ Delta(G)+2 and characterize graphs with equalities.;4. Star edge coloring. A star edge coloring of a graph is a proper edge coloring such that every connected 2-colored subgraph is a path with at most 3 edges. Let ch\u27st(G) be the list star chromatic index of G: the minimum s such that for every s-list assignment L for the edges, G has a star edge coloring from L. By introducing a stronger coloring, we show with a very concise proof that the upper bound of the star chromatic index of trees also holds for list star chromatic index of trees, i.e. ch\u27st( T) ≤ [3Delta/2] for any tree T with maximum degree Delta. And then by applying some orientation technique we present two upper bounds for list star chromatic index of k-degenerate graphs

Some problems in polynomial interpolation and topological complexity

This thesis is comprised of two projects in applied computational mathematics. In Chapter 1, we discuss the geometry and combinatorics of geometrically characterized sets. These are finite sets of n+d choose n points in R^d which impose independent conditions on polynomials of degree n, and which have Lagrange polynomials of a special form. These sets were introduced by Chung and Yao in a 1977 paper in the SIAM Journal of Numerical Analysis in the context of polynomial interpolation. There are several conjectures on the nature and geometric structure of these sets. We investigate the geometry and combinatorics of GC sets for d at least 2, and prove they are closely related to simplicial complexes which are Cohen-Macaulay and have a Cohen-Macaulay dual. In Chapter 2, we will discuss the motion planning problem in complex hyperplane arrangement complements. The difficulty of constructing a minimally discontinuous motion planning algorithm for a topological space X is measured by an integer invariant of X called topological complexity or TC(X). Yuzvinsky developed a combinatorial criterion for hyperplane arrangement complements which guarantees that their topological complexity is as large as possible. Applying this criterion in the special case when the arrangement is graphic, we simplify the criterion to an inequality on the edge density of the graph which is closely related to the inequality in the arboricity theorem of Nash-Williams

Running mass of the b-quark in QCD and SUSY QCD

The running mass of the b-quark defined in DRbar-scheme is one of the important parameters of SUSY QCD. To find its value it should be related to some known experimental input. In this paper the b-quark running mass defined in nonsupersymmetric QCD is chosen for determination of corresponding parameter in SUSY QCD. The relation between these two quantities is found by considering five-flavor QCD as an effective theory obtained from its supersymmetric extension. A numerical analysis of the calculated two-loop relation and its impact on the MSSM spectrum is discussed. Since for nonsupersymmetric models MSbar-scheme is more natural than DRbar, we also propose a new procedure that allows one to calculate relations between MSbar- and DRbar-parameters. Unphysical epsilon-scalars that give rise to the difference between mentioned schemes are assumed to be heavy and decoupled in the same way as physical degrees of freedom. By means of this method it is possible to ``catch two rabbits'', i.e., decouple heavy particles and turn from DRbar to MSbar, at the same time. Explicit two-loop example of DRbar -> MSbar transition is given in the context of QCD. The advantages and disadvantages of the method are briefly discussed.Comment: 33 pages, 6 figures, 1 table, typos corrected, added references

Infinite State AMC-Model Checking for Cryptographic Protocols

Only very little is known about the automatic analysis of cryptographic protocols for game-theoretic security properties. In this paper, we therefore study decidability and complexity of the model checking problem for AMC-formulas over infinite state concurrent game structures induced by cryptographic protocols and the Dolev-Yao intruder. We show that the problem is NEXPTIME-complete when making reasonable assumptions about protocols and for an expressive fragment of AMC, which contains, for example, all properties formulated by Kremer and Raskin in fair ATL for contract-signing and non-repudiation protocols. We also prove that our assumptions on protocols are necessary to obtain decidability

Components of domino tilings under flips in quadriculated cylinder and torus

In a region $R$ consisting of unit squares, a domino is the union of two adjacent squares and a (domino) tiling is a collection of dominoes with disjoint interior whose union is the region. The flip graph $\mathcal{T}(R)$ is defined on the set of all tilings of $R$ such that two tilings are adjacent if we change one to another by a flip (a $90^{\circ}$ rotation of a pair of side-by-side dominoes). It is well-known that $\mathcal{T}(R)$ is connected when $R$ is simply connected. By using graph theoretical approach, we show that the flip graph of $2m\times(2n+1)$ quadriculated cylinder is still connected, but the flip graph of $2m\times(2n+1)$ quadriculated torus is disconnected and consists of exactly two isomorphic components. For a tiling $t$, we associate an integer $f(t)$, forcing number, as the minimum number of dominoes in $t$ that is contained in no other tilings. As an application, we obtain that the forcing numbers of all tilings in $2m\times (2n+1)$ quadriculated cylinder and torus form respectively an integer interval whose maximum value is $(n+1)m$

Rethinking Tokenizer and Decoder in Masked Graph Modeling for Molecules

Masked graph modeling excels in the self-supervised representation learning of molecular graphs. Scrutinizing previous studies, we can reveal a common scheme consisting of three key components: (1) graph tokenizer, which breaks a molecular graph into smaller fragments (i.e., subgraphs) and converts them into tokens; (2) graph masking, which corrupts the graph with masks; (3) graph autoencoder, which first applies an encoder on the masked graph to generate the representations, and then employs a decoder on the representations to recover the tokens of the original graph. However, the previous MGM studies focus extensively on graph masking and encoder, while there is limited understanding of tokenizer and decoder. To bridge the gap, we first summarize popular molecule tokenizers at the granularity of node, edge, motif, and Graph Neural Networks (GNNs), and then examine their roles as the MGM's reconstruction targets. Further, we explore the potential of adopting an expressive decoder in MGM. Our results show that a subgraph-level tokenizer and a sufficiently expressive decoder with remask decoding have a large impact on the encoder's representation learning. Finally, we propose a novel MGM method SimSGT, featuring a Simple GNN-based Tokenizer (SGT) and an effective decoding strategy. We empirically validate that our method outperforms the existing molecule self-supervised learning methods. Our codes and checkpoints are available at https://github.com/syr-cn/SimSGT.Comment: NeurIPS 2023. 10 page