arXiv subjects
Publications and source records attributed to Robin Thomas.
We show that the size of a 4-critical graph of girth at least five is bounded by a linear function of its genus. This strengthens the previous bound on the size of such graphs given by Thomassen. It also serves as the basic case for the description of the structure of 4-critical triangle-free graphs embedded in a fixed surface, presented in a future paper of this series.
An embedding of a graph in $3$-space is linkless if for every two disjoint cycles there exists an embedded ball that contains one of the cycles and is disjoint from the other. We prove that every bipartite linklessly embeddable (simple) graph on $n\ge5$ vertices has at most $3n-10$ edges, unless it is isomorphic to the complete bipartite graph $K_{3,n-3}$.
We prove two results: 1. A graph $G$ on at least seven vertices with a vertex $v$ such that $G-v$ is planar and $t$ triangles satisfies $|E(G)| \leq 3|V(G)|- 9 + t/3$. 2. For $p=2,3,\ldots,9$, a triangle-free graph $G$ on at least $2p-5$ vertices with no $K_p$-minor satisfies $|E(G)|\leq (p-2)|V(G)| - (p-2)^2$.
Let $G$ be a graph embedded in a fixed surface $Σ$ of genus $g$ and let $L=(L(v):v\in V(G))$ be a collection of lists such that either each list has size at least five, or each list has size at least four and $G$ is triangle-free, or each list has size at least three and $G$ has no cycle of length four or less. An $L$-coloring of $G$ is a mapping $ϕ$ with domain $V(G)$ such that $ϕ(v)\in L(v)$ for every $v\in V(G)$ and $ϕ(v)\neϕ(u)$ for every pair of adjacent vertices $u,v\in V(G)$. We prove * if every non-null-homotopic cycle in $G$ has length $Ω(\log g)$, then $G$ has an $L$-coloring, * if $G$ does not have an $L$-coloring, but every proper subgraph does ("$L$-critical graph"), then $|V(G)|=O(g)$, * if every non-null-homotopic cycle in $G$ has length $Ω(g)$, and a set $X\subseteq V(G)$ of vertices that are pairwise at distance $Ω(1)$ is precolored from the corresponding lists, then the precoloring extends to an $L$-coloring of $G$, * if every non-null-homotopic cycle in $G$ has length $Ω(g)$, and the graph $G$ is allowed to have crossings, but every two crossings are at distance $Ω(1)$, then $G$ has an $L$-coloring, and * if $G$ has at least one $L$-coloring, then it has at least $2^{Ω(|V(G)|)}$ distinct $L$-colorings. We show that the above assertions are consequences of certain isoperimetric inequalities satisfied by $L$-critical graphs, and we study the structure of families of embedded graphs that satisfy those inequalities. It follows that the above assertions hold for other coloring problems, as long as the corresponding critical graphs satisfy the same inequalities.
Let $H$ be a fixed graph. What can be said about graphs $G$ that have no subgraph isomorphic to a subdivision of $H$? Grohe and Marx proved that such graphs $G$ satisfy a certain structure theorem that is not satisfied by graphs that contain a subdivision of a (larger) graph $H_1$. Dvořák found a clever strengthening---his structure is not satisfied by graphs that contain a subdivision of a graph $H_2$, where $H_2$ has "similar embedding properties" as $H$. Building upon Dvořák's theorem, we prove that said graphs $G$ satisfy a similar structure theorem. Our structure is not satisfied by graphs that contain a subdivision of a graph $H_3$ that has similar embedding properties as $H$ and has the same maximum degree as $H$. This will be important in a forthcoming application to well-quasi-ordering.
We prove that if a graph has a tree-decomposition of width at most w, then it has a tree-decomposition of width at most w with certain desirable properties. We will use this result in a subsequent paper to show that every 2-connected graph of large path-width has a minor isomorphic to either a large tree with a vertex attached to every vertex of the tree or a large outerplanar graph.
Let $P$ be a graph with a vertex $v$ such that $P\backslash v$ is a forest, and let $Q$ be an outerplanar graph. We prove that there exists a number $p=p(P,Q)$ such that every 2-connected graph of path-width at least $p$ has a minor isomorphic to $P$ or $Q$. This result answers a question of Seymour and implies a conjecture of Marshall and Wood. The proof is based on a new property of tree-decompositions.
We prove an ear-decomposition theorem for $4$-edge-connected graphs and use it to prove that for every $4$-edge-connected graph $G$ and every $r\in V(G)$, there is a set of four spanning trees of $G$ with the following property. For every vertex in $G$, the unique paths back to $r$ in each tree are edge-disjoint. Our proof implies a polynomial-time algorithm for constructing the trees.
Let G be a plane graph of girth at least five. We show that if there exists a 3-coloring phi of a cycle C of G that does not extend to a 3-coloring of G, then G has a subgraph H on O(|C|) vertices that also has no 3-coloring extending phi. This is asymptotically best possible and improves a previous bound of Thomassen. In the next paper of the series we will use this result and the attendant theory to prove a generalization to graphs on surfaces with several precolored cycles.
A cubic graph $G$ is cyclically 5-connected if $G$ is simple, 3-connected, has at least 10 vertices and for every set $F$ of edges of size at most four, at most one component of $G\backslash F$ contains circuits. We prove that if $G$ and $H$ are cyclically 5-connected cubic graphs and $H$ topologically contains $G$, then either $G$ and $H$ are isomorphic, or (modulo well-described exceptions) there exists a cyclically 5-connected cubic graph $G'$ such that $H$ topologically contains $G'$ and $G'$ is obtained from $G$ in one of the following two ways. Either $G'$ is obtained from $G$ by subdividing two distinct edges of $G$ and joining the two new vertices by an edge, or $G'$ is obtained from $G$ by subdividing each edge of a circuit of length five and joining the new vertices by a matching to a new circuit of length five disjoint from $G$ in such a way that the cyclic orders of the two circuits agree. We prove a companion result, where by slightly increasing the connectivity of $H$ we are able to eliminate the second construction. We also prove versions of both of these results when $G$ is almost cyclically 5-connected in the sense that it satisfies the definition except for 4-edge cuts such that one side is a circuit of length four. In this case $G'$ is required to be almost cyclically 5-connected and to have fewer circuits of length four than $G$. In particular, if $G$ has at most one circuit of length four, then $G'$ is required to be cyclically 5-connected. However, in this more general setting the operations describing the possible graphs $G'$ are more complicated.
A graph H is t-apex if H-X is planar for some subset X of V(H) of size t. For any integer t>=0 and a fixed t-apex graph H, we give a polynomial-time algorithm to decide whether a (t+3)-connected H-minor-free graph is colorable from a given assignment of lists of size t+4. The connectivity requirement is the best possible in the sense that for every t>=1, there exists a t-apex graph H such that testing (t+4)-colorability of (t+2)-connected H-minor-free graphs is NP-complete. Similarly, the size of the lists cannot be decreased (unless P=NP), since for every t>=1, testing (t+3)-list-colorability of (t+3)-connected K_{t+4}-minor-free graphs is NP-complete.
We give an elementary and self-contained proof, and a numerical improvement, of a weaker form of the excluded clique minor theorem of Robertson and Seymour, the following. Let t,r>0 be integers, and let R=49152t^{24}(40t^2+r). An r-wall is obtained from a (2r x r)-grid by deleting every odd vertical edge in every odd row and every even vertical edge in every even row, then deleting the two resulting vertices of degree one, and finally subdividing edges arbitrarily. The vertices of degree two that existed before the subdivision are called the pegs of the r-wall. Let G be a graph with no K_t minor, and let W be an R-wall in G. We prove that there exist a subset A of V(G) of size at most 12288t^{24} and an r-subwall W' of W such that V(W') is disjoint from A and W' is a flat wall in G-A in the following sense. There exists a separation (X,Y) of G-A such that X\cap Y is a subset of the vertex set of the cycle C' that bounds the outer face of W', V(W') is a subset of Y, every peg of W' belongs to X and the graph G[Y] can almost be drawn in the unit disk with the vertices X\cap Y drawn on the boundary of the disk in the order determined by C'. Here almost means that the assertion holds after repeatedly removing parts of the graph separated from X\cap Y by a cutset Z of size at most three, and adding all edges with both ends in Z. Our proof gives rise to an algorithm that runs in polynomial time even when r and t are part of the input instance. The proof is self-contained in the sense that it uses only results whose proofs can be found in textbooks.
Let $G$ be a plane graph with outer cycle $C$ and let $(L(v):v\in V(G))$ be a family of non-empty sets. By an $L$-coloring of $G$ we mean a (proper) coloring $ϕ$ of $G$ such that $ϕ(v)\in L(v)$ for every vertex $v$ of $G$. Thomassen proved that if $v_1,v_2\in V(C)$ are adjacent, $L(v_1)\ne L(v_2)$, $|L(v)|\ge3$ for every $v\in V(C)-\{v_1,v_2\}$ and $|L(v)|\ge5$ for every $v\in V(G)-V(C)$, then $G$ has an $L$-coloring. What happens when $v_1$ and $v_2$ are not adjacent? Then an $L$-coloring need not exist, but in the first paper of this series we have shown that it exists if $|L(v_1)|,|L(v_2)|\ge2$. Here we characterize when an $L$-coloring exists if $|L(v_1)|\ge1$ and $|L(v_2)|\ge2$. This result is a lemma toward a more general theorem along the same lines, which we will use to prove that minimally non-$L$-colorable planar graphs with two precolored cycles of bounded length are of bounded size. The latter result has a number of applications which we pursue elsewhere.
Gyori and Lovasz independently proved the following beautiful theorem. Let $k\ge2$ be an integer, let $G$ be a $k$-connected graph on $n$ vertices, let $v_1,v_2,\ldots,v_k$ be distinct vertices of $G$ and let $n_1,n_2,\ldots,n_k$ be positive integers with $n_1+n_2+\cdots+n_k=n$. Then $G$ has disjoint connected subgraphs $G_1,G_2,\ldots,G_k$ such that for $i=1,2,\ldots,k$ the graph $G_i$ has $n_i$ vertices and $v_i\in V(G_i)$. We give a self-contained exposition of Gyori's proof.
Let G be a plane graph with exactly one triangle T and all other cycles of length at least 5, and let C be a facial cycle of G of length at most six. We prove that a 3-coloring of C does not extend to a 3-coloring of G if and only if C has length exactly six and there is a color x such that either G has an edge joining two vertices of C colored x, or T is disjoint from C and every vertex of T is adjacent to a vertex of C colored x. This is a lemma to be used in a future paper of this series.
A graph is apex if there is a vertex whose deletion makes the graph planar, and doublecross if it can be drawn in the plane with only two crossings, both incident with the infinite region in the natural sense. In 1966, Tutte conjectured that every two-edge-connected cubic graph with no Petersen graph minor is three-edge-colourable. With Neil Robertson, two of us showed that this is true in general if it is true for apex graphs and doublecross graphs. In another paper, two of us solved the apex case, but the doublecross case remained open. Here we solve the doublecross case; that is, we prove that every two-edge-connected doublecross cubic graph is three-edge-colourable. The proof method is a variant on the proof of the four-colour theorem.
In [Directed tree-width, J. Combin. Theory Ser. B 82 (2001), 138-154] we introduced the notion of tree-width of directed graphs and presented a conjecture, formulated during discussions with Noga Alon and Bruce Reed, stating that a digraph of huge tree-width has a large "cylindrical grid" minor. Here we prove the conjecture for planar digraphs, but many steps of the proof work in general. This is an unedited and unpolished manuscript from October 2001. Since many people asked for copies we are making it available in the hope that it may be useful. The conjecture was proved by Kawarabayashi and Kreutzer in arXiv:1411.5681.
Let $G$ be a plane graph with outer cycle $C$ and let $(L(v):v\in V(G))$ be a family of sets such that $|L(v)|\ge 5$ for every $v\in V(G)$. By an $L$-coloring of a subgraph $J$ of $G$ we mean a (proper) coloring $ϕ$ of $J$ such that $ϕ(v)\in L(v)$ for every vertex $v$ of $J$. We prove a conjecture of Dvorak et al. that if $H$ is a minimal subgraph of $G$ such that $C$ is a subgraph of $H$ and every $L$-coloring of $C$ that extends to an $L$-coloring of $H$ also extends to an $L$-coloring of $G$, then $|V(H)|\le 19|V(C)|$. This is a lemma that plays an important role in subsequent papers, because it motivates the study of graphs embedded in surfaces that satisfy an isoperimetric inequality suggested by this result. Such study turned out to be quite profitable for the subject of list coloring graphs on surfaces.