32,585 research outputs found
Filling the complexity gaps for colouring planar and bounded degree graphs.
We consider a natural restriction of the List Colouring problem, k-Regular List Colouring, which corresponds to the List Colouring problem where every list has size exactly k. We give a complete classification of the complexity of k-Regular List Colouring restricted to planar graphs, planar bipartite graphs, planar triangle-free graphs and to planar graphs with no 4-cycles and no 5-cycles. We also give a complete classification of the complexity of this problem and a number of related colouring problems for graphs with bounded maximum degree
Filling the complexity gaps for colouring planar and bounded degree graphs
We consider a natural restriction of the List Colouring problem, k-Regular List Colouring, which corresponds to the List Colouring problem where every list has size exactly k. We give a complete classification of the complexity of k-Regular List Colouring restricted to planar graphs, planar bipartite graphs, planar triangle-free graphs and to planar graphs with no 4-cycles and no 5-cycles. We also give a complete classification of the complexity of this problem and a number of related colouring problems for graphs with bounded maximum degree
On equality in an upper bound for the acyclic domination number
A subset of vertices in a graph is acyclic if the subgraph it induces contains no cycles. The acyclic domination number of a graph is the minimum cardinality of an acyclic dominating set of . For any graph with vertices and maximum degree , . In this paper we characterize the connected graphs and the connected triangle-free graphs which achieve this upper bound
Fine Structure of 4-Critical Triangle-Free Graphs II. Planar Triangle-Free Graphs with Two Precolored 4-Cycles
We study 3-coloring properties of triangle-free planar graphs with two precolored 4-cycles and that are far apart. We prove that either every precoloring of extends to a 3-coloring of , or contains one of two special substructures which uniquely determine which 3-colorings of extend. As a corollary, we prove that there exists a constant such that if is a planar triangle-free graph and if consists of vertices at pairwise distances at least , then every precoloring of extends to a 3-coloring of . This gives a positive answer to a conjecture of Dvořák, Král\u27, and Thomas, and implies an exponential lower bound on the number of 3-colorings of triangle-free planar graphs of bounded maximum degree
Graphs with few Hamiltonian Cycles
We describe an algorithm for the exhaustive generation of non-isomorphic
graphs with a given number of hamiltonian cycles, which is especially
efficient for small . Our main findings, combining applications of this
algorithm and existing algorithms with new theoretical results, revolve around
graphs containing exactly one hamiltonian cycle (1H) or exactly three
hamiltonian cycles (3H). Motivated by a classic result of Smith and recent work
of Royle, we show that there exist nearly cubic 1H graphs of order iff is even. This gives the strongest form of a theorem of Entringer and
Swart, and sheds light on a question of Fleischner originally settled by
Seamone. We prove equivalent formulations of the conjecture of Bondy and
Jackson that every planar 1H graph contains two vertices of degree 2, verify it
up to order 16, and show that its toric analogue does not hold. We treat
Thomassen's conjecture that every hamiltonian graph of minimum degree at least
contains an edge such that both its removal and its contraction yield
hamiltonian graphs. We also verify up to order 21 the conjecture of Sheehan
that there is no 4-regular 1H graph. Extending work of Schwenk, we describe all
orders for which cubic 3H triangle-free graphs exist. We verify up to order
Cantoni's conjecture that every planar cubic 3H graph contains a triangle,
and show that there exist infinitely many planar cyclically 4-edge-connected
cubic graphs with exactly four hamiltonian cycles, thereby answering a question
of Chia and Thomassen. Finally, complementing work of Sheehan on 1H graphs of
maximum size, we determine the maximum size of graphs containing exactly one
hamiltonian path and give, for every order , the exact number of such graphs
on vertices and of maximum size.Comment: 29 pages; to appear in Mathematics of Computatio
Structure and algorithms for (cap, even hole)-free graphs
A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free graphs, and more generally, (cap, 4-hole)-free odd-signable graphs. We give an explicit construction of these graphs. We prove that every such graph G has a vertex of degree at most [View the MathML source], and hence [View the MathML source], where ω(G) denotes the size of a largest clique in G and χ(G) denotes the chromatic number of G. We give an O(nm) algorithm for q-coloring these graphs for fixed q and an O(nm) algorithm for maximum weight stable set, where n is the number of vertices and m is the number of edges of the input graph. We also give a polynomial-time algorithm for minimum coloring. Our algorithms are based on our results that triangle-free odd-signable graphs have treewidth at most 5 and thus have clique-width at most 48, and that (cap, 4-hole)-free odd-signable graphs G without clique cutsets have treewidth at most 6ω(G)−1 and clique-width at most 48
Extremal problems involving forbidden subgraphs
In this thesis, we study extremal problems involving forbidden subgraphs. We are interested in extremal problems over a family of graphs or over a family of hypergraphs.
In Chapter 2, we consider improper coloring of graphs without short cycles. We find how sparse an improperly critical graph can be when it has no short cycle. In particular, we find the exact threshold of density of triangle-free -colorable graphs and we find the asymptotic threshold of density of -colorable graphs of large girth when .
In Chapter 3, we consider other variations of graph coloring. We determine harmonious chromatic number of trees with large maximum degree and show upper bounds of -dynamic chromatic number of graphs in terms of other parameters.
In Chapter 4, we consider how dense a hypergraph can be when we forbid some subgraphs.
In particular, we characterize hypergraphs with the maximum number of edges that contain no -regular subgraphs. We also establish upper bounds for the number of edges in graphs and hypergraphs with no edge-disjoint equicovering subgraphs
- …