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Alternating cycles and paths in edgecoloured multigraphs: a survey
, 1995
"... A path or cycle in an edgecoloured multigraph is called alternating if its successive edges differ in colour. We survey results of both theoretical and algorithmic character concerning alternating cycles and paths in edgecoloured multigraphs. We also show useful connections between the theory of p ..."
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Cited by 11 (5 self)
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A path or cycle in an edgecoloured multigraph is called alternating if its successive edges differ in colour. We survey results of both theoretical and algorithmic character concerning alternating cycles and paths in edgecoloured multigraphs. We also show useful connections between the theory of paths and cycles in bipartite digraphs and the theory of alternating paths and cycles in edgecoloured graphs.
Proofs Without Syntax
 Annals of Mathematics
"... [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 combinatori ..."
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Cited by 5 (0 self)
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[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 (graphtheoretic), 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.
On Theorems Equivalent with Kotzig's Result on Graphs with Unique 1Factors
, 2001
"... We show that several known theorems on graphs and digraphs are equivalent. The list of equivalent theorems include Kotzig's result on graphs with unique 1factors, a lemma by Seymour and Giles, theorems on alternating cycles in edgecolored graphs, and a theorem on semicycles in digraphs. We co ..."
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Cited by 1 (1 self)
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We show that several known theorems on graphs and digraphs are equivalent. The list of equivalent theorems include Kotzig's result on graphs with unique 1factors, a lemma by Seymour and Giles, theorems on alternating cycles in edgecolored graphs, and a theorem on semicycles in digraphs. We consider computational problems related to the quoted results; all these problems ask whether a given (di)graph contains a cycle satisfying certain properties which runs through p prescribed vertices. We show that all considered problems can be solved in polynomial time for p < 2 but are NPcomplete for p # 2. 1