Results 1 
5 of
5
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 ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
[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. ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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
APPROVAL
, 1993
"... ii A graph is said to be kextendable if it is matchable and every matching of size k extends to a perfect matching. The notion of extendability has been studied by a number of authors. Most highly extendable graphs that have appeared in the literature have high edge density. Indeed, it is a nontriv ..."
Abstract
 Add to MetaCart
ii A graph is said to be kextendable if it is matchable and every matching of size k extends to a perfect matching. The notion of extendability has been studied by a number of authors. Most highly extendable graphs that have appeared in the literature have high edge density. Indeed, it is a nontrivial problem to find graphs that have high extendability and whose girth is at least five. In light of these facts it seems to be interesting to look for constructions yielding graphs without short cycles whose extendability would also be large. This is in focus of this thesis the main result of which is a constructive proof of the existence of highly extendable graphs whose girth is greater than a given number. iii Acknowledgements I am thankful to professor Jaroslav Neˇsetˇril for introducing me to the problem of constructing highly extendable graphs without short cycles as well as for advising me
Deciding the deterministic property for soliton graphs
"... Soliton automata are a mathematical model for electronic switching at the molecular level. In the design of soliton circuits, deterministic automata are of primary importance. The underlying graphs of such automata, called soliton grahs, are characterized in terms of generalized trees and graphs hav ..."
Abstract
 Add to MetaCart
Soliton automata are a mathematical model for electronic switching at the molecular level. In the design of soliton circuits, deterministic automata are of primary importance. The underlying graphs of such automata, called soliton grahs, are characterized in terms of generalized trees and graphs having a unique perfect matching. Based on this characterization, a modification of the currently most efficient unique perfect matching algorithm is worked out to decide in O(m log 4 n) time if a graph with n vertices and m edges defines a deterministic soliton automaton. A yet more efficient O(m) algorithm is given for the special case of chestnut and elementary soliton graphs. All of these algorithms are capable of constructing a state for the corresponding soliton automaton found, and the general algorithm can also be used to simplify the automaton to an isomorphic elementary one. 1.