A sequence S is potentially Km −C4-graphical if it has a realization containing a Km − C4 as a subgraph. Let σ(Km − C4, n) denote the smallest degree sum such that every n-term graphical sequence S with σ(S) ≥ σ(Km − C4, n) is potentially Km − C4-graphical. In this paper, we prove that σ(Km − C4, n) ≥ (2m − 6)n − (m − 3)(m − 2) + 2, for n ≥ m ≥ 4. We conjecture that equality holds for n ≥ m ≥ 4. We prove that this conjecture is true for m = 5. Key words: graph; degree sequence; potentially Km − C4-graphic sequence AMS Subject Classifications: 05C07, 05C35 1

Abstract. In this paper we consider the question of whether NC 0 circuits can generate pseudorandom distributions. While we leave the general question unanswered, we show • Generators computed by NC 0 circuits where each output bit depends on at most 3 input bits (i.e, NC 0 3 circuits) and with stretch factor greater than 4 are not pseudorandom. • A large class of “non-problematic ” NC 0 generators with superlinear stretch (including all NC 0 3 generators with superlinear stretch) are broken by a statistical test based on a linear dependency test combined with a pairwise independence test. • There is an NC 0 4 generator with a super-linear stretch that passes the linear dependency test as well as k-wise independence tests, for any constant k. 1

A sequence S is potentially Km −Pk graphical if it has a realization containing a Km −Pk as a subgraph. Let σ(Km −Pk, n) denote the smallest degree sum such that every n-term graphical sequence S with σ(S) ≥ σ(Km−Pk, n) is potentially Km−Pk graphical. In this paper, we prove that σ(Km−Pk, n) ≥ (2m−6)n−(m−3)(m−2)+2, for n ≥ m ≥ k + 1 ≥ 4. We conjectured that equality holds for n ≥ m ≥ k + 1 ≥ 4. We proved that this conjecture is true for m = k + 1 = 5 and m = k + 2 = 5. Key words: graph; degree sequence; potentially Km − Pk-graphic sequence

Let Kk, Ck, Tk, and Pk denote a complete graph on k vertices, a cycle on k vertices, a tree on k + 1 vertices, and a path on k + 1 vertices, respectively. Let Km −H be the graph obtained from Km by removing the edges set E(H) of the graph H (H is a subgraph of Km). A sequence S is potentially Km − H-graphical if it has a realization containing a Km − H as a subgraph. Let σ(Km − H, n) denote the smallest degree sum such that every n-term graphical sequence S with σ(S) ≥ σ(Km − H, n) is potentially Km − H-graphical. In this paper, we determine the values of σ(Kr+1−H, n) for n ≥ 4r+10, r ≥ 3, r+1 ≥ k ≥ 4 where H is a graph on k vertices which contains a tree on 4 vertices but not contains a cycle on 3 vertices. We also determine the values of σ(Kr+1 − P2, n) for n ≥ 4r + 8, r ≥ 3. Key words: graph; degree sequence; potentially Kr+1 − H-graphic sequence AMS Subject Classifications: 05C07, 05C35 1

[M]athematicians care no more for logic than logicians for mathematics. Augustus de Morgan, 1868 Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatorial (graph-theoretic), rather than syntactic. It defines a combinatorial proof of a proposition φ as a graph homomorphism h: C → G(φ), where G(φ) is a graph associated with φ and C is a coloured graph. The main theorem is soundness and completeness: φ is true if and only if there exists a combinatorial proof h: C → G(φ). 1.

Abstract. In this paper we consider a variation of the classical extremal problems. Let S be a n-term graphical sequence, and σ(S) be the sum of the terms in S. Let G be a graph. The problem is to determine the smallest even m such that any n-term graphical sequence S having σ(S) ≥ m has a realization containing G. Denote this value by σ(G, n). We show σ(C2m+1, n) = m(2n − m − 1) + 2, for m ≥ 3, n ≥ 3m; σ(C2m+2, n) = m(2n − m − 1) + 4, for m ≥ 3, n ≥ 5m − 2. Key words. graph, degree sequence, potentially G-graphic sequence AMS subject classification. 05C35

Abstract—Hierarchical high-level PNs (HHPNs) with time versions are a useful tool to model systems in a variety of application domains, ranging from logistics to complex workflows. This paper addresses an application domain which is receiving more and more attention: procedure that arranges the final inpatient charge in payment’s office and their management. We shall prove that Petri net based analysis is able to improve the delays during the procedure, in order that inpatient charges could be more reliable and on time.

A sequence S is potentially Kp,1,1 graphical if it has a realization containing a Kp,1,1 as a subgraph, where Kp,1,1 is a complete 3-partite graph with partition sizes p, 1, 1. Let σ(Kp,1,1, n) denote the smallest degree sum such that every n-term graphical sequence S with σ(S) ≥ σ(Kp,1,1, n) is potentially Kp,1,1 graphical. In this paper, we prove that σ(Kp,1,1, n) ≥ 2[((p + 1)(n − 1) + 2)/2] for n ≥ p + 2. We conjecture that equality holds for n ≥ 2p + 4. We prove that this conjecture is true for p = 3.

A sequence S is potentially Km − H-graphical if it has a realization containing a Km − H as a subgraph. Let σ(Km − H, n) denote the smallest degree sum such that every n-term graphical sequence S with σ(S) ≥ σ(Km −H, n) is potentially Km−H-graphical. In this paper, we determine σ(Kr+1−(kP2 tK2), n) for n ≥ 4r+10, r+1 ≥ 3k+2t, k+t ≥ 2, k ≥ 1, t ≥ 0. Key words: graph; degree sequence; potentially Kr+1 − (kP2 tK2)graphic sequence AMS Subject Classifications: 05C07, 05C35 1

-graphic sequences