最小度不小于3的图的圈长问题

Bondy和Vince曾证明最小度不小于3的图包含两个长度相差为1或者2的圈,这个结果回答了ErdSs提出的问题．Haggkvist和scott证明了除肠外,所有的3-正则图都包含两个长度相差2的圈．通过不同的方法,我们得到了下面的结论：除了每个端块都是硒的图外,所有最小度不小于3的图都包含两个长度相差2的圈．     

Cayley图的泛因子性质

一个κ-正则图若满足对任意正整数s,1≤s≤κ,均存在一个s-因子或一个2[s/2]因子,则称其有泛因子或偶泛因子性质.本文证明了每个奇度Cayley图是泛因子的,每个偶度Carley图是偶泛因子的.同时证明了二面体群上的每个Cayley图均是泛因子的.     

Coloring of Plane Graphs with Unique Maximal Colors on Faces

The Four Color Theorem asserts that the vertices of every plane graph can be properly colored with four colors. Fabrici and Göring conjectured the following stronger statement to also hold: the vertices of every plane graph can be properly colored with the numbers 1, …, 4 in such a way that every face contains a unique vertex colored with the maximal color appearing on that face. They proved that every plane graph has such a coloring with the numbers 1, …, 6. We prove that every plane graph has such a coloring with the numbers 1, …, 5 and we also prove the list variant of the statement for lists of sizes seven.     

Cubic vertices in planar hypohamiltonian graphs

Thomassen showed in 1978 that every planar hypohamiltonian graph contains a cubic vertex. Equivalently, a planar graph with minimum degree at least 4 in which every vertex-deleted subgraph is hamiltonian, must be itself hamiltonian. By applying work of Brinkmann and the author, we extend this result in three directions. We prove that (i) every planar hypohamiltonian graph contains at least four cubic vertices, (ii) every planar almost hypohamiltonian graph contains a cubic vertex, which is not the exceptional vertex (solving a problem of the author raised in J. Graph Theory [79 (2015) 63–81]), and (iii) every hypohamiltonian graph with crossing number 1 contains a cubic vertex. Furthermore, we settle a recent question of Thomassen by proving that asymptotically the ratio of the minimum number of cubic vertices to the order of a planar hypohamiltonian graph vanishes.     

Line graphs of multigraphs and Hamilton‐connectedness of claw‐free graphs

We introduce a closure concept that turns a claw‐free graph into the line graph of a multigraph while preserving its (non‐)Hamilton‐connectedness. As an application, we show that every 7‐connected claw‐free graph is Hamilton‐connected, and we show that the well‐known conjecture by Matthews and Sumner (every 4‐connected claw‐free graph is hamiltonian) is equivalent with the statement that every 4‐connected claw‐free graph is Hamilton‐connected. Finally, we show a natural way to avoid the non‐uniqueness of a preimage of a line graph of a multigraph, and we prove that the closure operation is, in a sense, best possible. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:152‐173, 2011     

Contractible Subgraphs,Thomassen's Conjecture and the Dominating Cycle Conjecture for Snarks

We show that the conjectures by Matthews and Sumner (every 4-connected claw-free graph is hamiltonian), by Thomassen (every 4-connected line graph is hamiltonian) and by Fleischner (every cyclically 4-edge-connected cubic graph has either a 3-edge-coloring or a dominating cycle), which are known to be equivalent, are equivalent with the statement that every snark (i.e. a cyclically 4-edge-connected cubic graph of girth at least five that is not 3-edge-colorable) has a dominating cycle.We use a refinement of the contractibility technique which was introduced by Ryjáček and Schelp in 2003 as a common generalization and strengthening of the reduction techniques by Catlin and Veldman and of the closure concept introduced by Ryjáček in 1997.     

Every connected regular graph of even degree is a Schreier coset graph

Using Petersen's theorem, that every regular graph of even degree is 2-factorable, it is proved that every connected regular graph of even degree is isomorphic to a Schreier coset graph. The method used is a special application of the permutation voltage graph construction developed by the author and Tucker. This work is related to graph imbedding theory, because a Schreier coset graph is a covering space of a bouquet of circles.     

The Hadwiger number of infinite vertex-transitive graphs

We prove that every infinite, locally finite 3-connected, almost 4-connected, almost transitive, nonplanar graph, which contains infinitely many pairwise disjoint infinite paths belonging to the same end, can be contracted into an infinite complete graph. This implies that every infinite, locally finite, connected, nonplanar vertex-transitive graph with only one end can be contracted into an infinite complete graph. This problem was raised by L. Babai.     

Edge Partition of Graphs Embeddable in the Projective Plane and the Klein Bottle

In a previous paper by the author joint with Baogang XU published in Discrete Math in 2018, we show that every non-planar toroidal graph can be edge partitioned into a planar graph and an outerplanar graph. This edge partition then implies some results in thickness and outerthickness of toroidal graphs. In particular, if each planar graph has outerthickness at most $2$ (conjectured by Chartrand, Geller and Hedetniemi in 1971 and the confirmation of the conjecture was announced by Gon\c{c}alves in 2005), then the outerthickness of toroidal graphs is at most 3 which is the best possible due to $K_7$. In this paper we continue to study the edge partition for projective planar graphs and Klein bottle embeddable graphs. We show that (1) every non-planar but projective planar graph can be edge partitioned into a planar graph and a union of caterpillar trees; and (2) every non-planar Klein bottle embeddable graph can be edge partitioned into a planar graph and a subgraph of two vertex amalgamation of a caterpillar tree with a cycle with pendant edges. As consequences, the thinkness of projective planar graphs and Klein bottle embeddabe graphs are at most $2$, which are the best possible, and the outerthickness of these graphs are at most $3$.     

Equistable graphs, general partition graphs, triangle graphs, and graph products

In this paper we examine the connections between equistable graphs, general partition graphs and triangle graphs. While every general partition graph is equistable and every equistable graph is a triangle graph, not every triangle graph is equistable, and a conjecture due to Jim Orlin states that every equistable graph is a general partition graph. The conjecture holds within the class of chordal graphs; if true in general, it would provide a combinatorial characterization of equistable graphs.Exploiting the combinatorial features of triangle graphs and general partition graphs, we verify Orlin’s conjecture for several graph classes, including AT-free graphs and various product graphs. More specifically, we obtain a complete characterization of the equistable graphs that are non-prime with respect to the Cartesian or the tensor product, and provide some necessary and sufficient conditions for the equistability of strong, lexicographic and deleted lexicographic products. We also show that the general partition graphs are not closed under the strong product, answering a question by McAvaney et al.     

On hamiltonian properties of 2-tough graphs

A well-known conjecture in hamiltonian graph theory states that every 2-tough graph is hamiltonian. We give some equivalent conjectures, e.g., the conjecture that every 2-tough graph is hamiltonian-connected.     

Contractible subgraphs, Thomassen’s conjecture and the dominating cycle conjecture for snarks

We show that the conjectures by Matthews and Sumner (every 4-connected claw-free graph is Hamiltonian), by Thomassen (every 4-connected line graph is Hamiltonian) and by Fleischner (every cyclically 4-edge-connected cubic graph has either a 3-edge-coloring or a dominating cycle), which are known to be equivalent, are equivalent to the statement that every snark (i.e. a cyclically 4-edge-connected cubic graph of girth at least five that is not 3-edge-colorable) has a dominating cycle.We use a refinement of the contractibility technique which was introduced by Ryjá?ek and Schelp in 2003 as a common generalization and strengthening of the reduction techniques by Catlin and Veldman and of the closure concept introduced by Ryjá?ek in 1997.     

On permutation labeling

We determine all permutation graphs of order ?9. We prove that every bipartite graph of order ?50 is a permutation graph. We convert the conjecture stating that “every tree is a permutation graph” to be “every bipartite graph is a permutation graph”.     

Minimal comparability completions of arbitrary graphs

A transitive orientation of an undirected graph is an assignment of directions to its edges so that these directed edges represent a transitive relation between the vertices of the graph. Not every graph has a transitive orientation, but every graph can be turned into a graph that has a transitive orientation, by adding edges. We study the problem of adding an inclusion minimal set of edges to an arbitrary graph so that the resulting graph is transitively orientable. We show that this problem can be solved in polynomial time, and we give a surprisingly simple algorithm for it. We use a vertex incremental approach in this algorithm, and we also give a more general result that describes graph classes Π for which Π completion of arbitrary graphs can be achieved through such a vertex incremental approach.     

Characterization of reducible hexagons and fast decomposition of elementary benzenoid graphs

A benzenoid graph is a finite connected plane graph with no cut vertices in which every interior region is bounded by a regular hexagon of a side length one. A benzenoid graph G is elementary if every edge belongs to a 1-factor of G. A hexagon h of an elementary benzenoid graph is reducible, if the removal of boundary edges and vertices of h results in an elementary benzenoid graph. We characterize the reducible hexagons of an elementary benzenoid graph. The characterization is the basis for an algorithm which finds the sequence of reducible hexagons that decompose a graph of this class in O(n²) time. Moreover, we present an algorithm which decomposes an elementary benzenoid graph with at most one pericondensed component in linear time.     

Parity and disparity subgraphs

A parity subgraph of a graph is a spanning subgraph such that the degrees of each vertex have the same parity in both the subgraph and the original graph. Known results include that every graph has an odd number of minimal parity subgraphs. Define a disparity subgraph to be a spanning subgraph such that each vertex has degrees of opposite parities in the subgraph and the original graph. (Only graphs with all even-order components can have disparity subgraphs). Every even-order spanning tree contains both a unique parity subgraph and a unique disparity subgraph. Moreover, every minimal disparity subgraph is shown to be paired by sharing a spanning tree with an odd number of minimal parity subgraphs, and every minimal parity subgraph is similarly paired with either one or an even number of minimal disparity subgraphs.     

Hamiltonicity of 3-connected line graphs

Thomassen conjectured that every 4-connected line graph is Hamiltonian. Lai et al. conjectured [H. Lai, Y. Shao, H. Wu, J. Zhou, Every 3-connected, essentially 11-connected line graph is Hamiltonian, J. Combin. Theory Ser. B 96 (2006) 571–576] that every 3-connected, essentially 4-connected line graph is Hamiltonian. In this note, we first show that the conjecture posed by Lai et al. is not true and there is an infinite family of counterexamples; we show that 3-connected, essentially 4-connected line graph of a graph with at most 9 vertices of degree 3 is Hamiltonian; examples show that all conditions are sharp.     

List 2-facial 5-colorability of plane graphs with girth at least 12

A proper vertex coloring of a plane graph is 2-facial if any two different vertices joined by a facial walk of length 2 are colored differently, and it is 2-distance if every two vertices at distance 2 from each other are colored differently. Note that any 2-facial coloring of a subcubic graph is 2-distance.It is known that every plane graph with girth at least 14 has a 2-facial 5-coloring [M. Montassier, A. Raspaud, A note on 2-facial coloring of plane graphs. Inform. Process. Lett. 98 (6) (2006) 235–241], and that every planar subcubic graph with girth at least 13 has a list 2-distance 5-coloring [F. Havet, Choosability of square of planar subcubic graphs with large girth, Discrete Math. 309 (2009) 3353–3563].We strengthen these results by proving the list 2-facial 5-colorability of plane graphs with girth at least 12.     

List-Coloring Claw-Free Graphs with Small Clique Number

Chudnovsky and Seymour proved that every connected claw-free graph that contains a stable set of size 3 has chromatic number at most twice its clique number. We improve this for small clique size, showing that every claw-free graph with clique number at most 3 is 4-choosable and every claw-free graph with clique number at most 4 is 7-choosable. These bounds are tight.     

Discs in unbreakable graphs

We show that every vertex in an unbreakable graph is in a disc, where a disc is a chordless cycle, or the complement of a chordless cycle, with at least five vertices. A corollary is that every vertex in a minimal imperfect graph is in a disc.This research was supported by NSERC Operating Grant OGP-0137764.