Graphs Of Girth Research Articles

Planar graphs of girth at least five are square (Δ + 2)-choosable

We prove a conjecture of Dvořák, Král, Nejedlý, and Škrekovski that planar graphs of girth at least five with maximum degree Δ are square (Δ+2)-colorable for large enough Δ. In fact, we prove the stronger statement that such graphs are square (Δ+2)-choosable and even square (Δ+2)-paintable.

The list chromatic number of graphs with small clique number

We prove that every triangle-free graph with maximum degree Δ has list chromatic number at most (1+o(1))Δln⁡Δ. This matches the best-known upper bound for graphs of girth at least 5. We also provide a new proof that for any r≥4 every Kr-free graph has list-chromatic number at most 200rΔln⁡ln⁡Δln⁡Δ.

Tight bounds on the coefficients of partition functions via stability

Partition functions arise in statistical physics and probability theory as the normalizing constant of Gibbs measures and in combinatorics and graph theory as graph polynomials. For instance the partition functions of the hard-core model and monomer-dimer model are the independence and matching polynomials respectively.We show how stability results follow naturally from the recently developed occupancy method for maximizing and minimizing physical observables over classes of regular graphs, and then show these stability results can be used to obtain tight extremal bounds on the individual coefficients of the corresponding partition functions.As applications, we prove new bounds on the number of independent sets and matchings of a given size in regular graphs. For large enough graphs and almost all sizes, the bounds are tight and confirm the Upper Matching Conjecture of Friedland, Krop, and Markström and a conjecture of Kahn on independent sets for a wide range of parameters. Additionally we prove tight bounds on the number of q-colorings of cubic graphs with a given number of monochromatic edges, and tight bounds on the number of independent sets of a given size in cubic graphs of girth at least 5.

Decomposing C4‐free graphs under degree constraints

AbstractA celebrated theorem of Stiebitz 13 asserts that any graph with minimum degree at least can be partitioned into two parts that induce two subgraphs with minimum degree at least s and t, respectively. This resolved a conjecture of Thomassen. In this article, we prove that for , if a graph G contains no cycle of length four and has minimum degree at least , then G can be partitioned into two parts that induce two subgraphs with minimum degree at least s and t, respectively. This improves the result of Diwan in 5, where he proved the same statement for graphs of girth at least five. Our proof also works for the case of variable functions, in which the bounds are sharp as showing by some polarity graphs.

Rainbow connectivity of Moore cages of girth 6

Let G be an edge-colored graph. A path P of G is said to be rainbow if no two edges of P have the same color. An edge-coloring of G is a rainbowt-coloring if for any two distinct vertices u and v of G there are at least t internally vertex-disjoint rainbow (u,v)-paths. The rainbowt-connectivityrct(G) of a graph G is the minimum integer j such that there exists a rainbow t-coloring using j colors. A (k;g)-cage is a k-regular graph of girth g and minimum number of vertices denoted n(k;g). In this paper we focus on g=6. It is known that n(k;6)≥2(k2−k+1) and when n(k;6)=2(k2−k+1) the (k;6)-cage is called a Moore cage. In this paper we prove that the rainbow k-connectivity of a Moore (k;6)-cage G satisfies that k≤rck(G)≤k2−k+1. It is also proved that the rainbow 3-connectivity of the Heawood graph is 6 or 7.

Splitting a planar graph of girth 5 into two forests with trees of small diameter

In 1985, Mihok and recently Axenovich, Ueckerdt, and Weiner asked about the minimum integer g∗>3 such that every planar graph with girth at least g∗ admits a 2-colouring of its vertices where the length of every monochromatic path is bounded from above by a constant. By results of Glebov and Zambalaeva and of Axenovich et al., it follows that 5≤g∗≤6. In this paper we establish that g∗=5. Moreover, we prove that every planar graph of girth at least 5 admits a 2-colouring of its vertices such that every monochromatic component is a tree of diameter at most 6. We also present the list version of our result.

Size of the largest induced forest in subcubic graphs of girth at least four and five

AbstractIn this article, we address the maximum number of vertices of induced forests in subcubic graphs with girth at least four or five. We provide a unified approach to prove that every 2‐connected subcubic graph on n vertices and m edges with girth at least four or five, respectively, has an induced forest on at least or vertices, respectively, except for finitely many exceptional graphs. Our results improve a result of Liu and Zhao and are tight in the sense that the bounds are attained by infinitely many 2‐connected graphs. Equivalently, we prove that such graphs admit feedback vertex sets with size at most or , respectively. Those exceptional graphs will be explicitly constructed, and our result can be easily modified to drop the 2‐connectivity requirement.

Three-coloring triangle-free graphs on surfaces II. 4-critical graphs in a disk

Let G be a plane graph of girth at least five. We show that if there exists a 3-coloring ϕ 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 ϕ. 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.

On regular induced subgraphs of generalized polygons

The cage problem asks for the smallest number c(k,g) of vertices in a k-regular graph of girth g and graphs meeting this bound are known as cages. While cages are known to exist for all integers k⩾2 and g⩾3, the exact value of c(k,g) is known only for some small values of k,g and three infinite families where g∈{6,8,12} and k−1 is a prime power. These infinite families come from the incidence graphs of generalized polygons. Some of the best known upper bounds on c(k,g) for g∈{6,8,12} have been obtained by constructing small regular induced subgraphs of these cages.In this paper, we first use the Expander Mixing Lemma to give a general lower bound on the size of an induced k-regular subgraph of a regular bipartite graph in terms of the second largest eigenvalue of the host graph. We use this bound to show that the known construction of (k,6)-graphs using Baer subplanes of the Desarguesian projective plane is the best possible. For generalized quadrangles and hexagons, our bounds are new. In particular, we improve the known lower bound on the size of an induced q-regular subgraph of the classical generalized quadrangle Q(4,q) and show that the known constructions are asymptotically sharp, which answers a question of Metsch [21, Section 6].For prime powers q, we also improve the known upper bounds on c(q,8) and c(q,12) by giving new geometric constructions of q-regular induced subgraphs in the symplectic generalized quadrangle W(3,q) and the split Cayley hexagon H(q), respectively. Our constructions show thatc(q,8)⩽2(q3−qq−q) for q an even power of a prime, andc(q,12)⩽2(q5−3q3) for all prime powers q. For q∈{3,4,5} we also give a computer classification of all q-regular induced subgraphs of the classical generalized quadrangles of order q. For W(3,7) we classify all 7-regular induced subgraphs which have a non-trivial automorphism.

The Game Coloring Number of Planar Graphs with a Specific Girth

Let \(\mathrm{col_g}(G)\) be the game coloring number of a given graph G. Define the game coloring number of a family of graphs \(\mathcal {H}\) as \(\mathrm{col_g}(\mathcal {H}) := \max \{\mathrm{col_g}(G):G \in \mathcal {H}\}.\) Let \(\mathcal {P}_k\) be the family of planar graphs of girth at least k. We show that \(\mathrm{col_g}(\mathcal {P}_7) \le 5.\) This result extends a result about the game coloring number by Wang and Zhang [10] (\(\mathrm{col_g}(\mathcal {P}_8) \le 5).\) We also show that these bounds are sharp by constructing a graph G where \(G \in \mathcal {P}_k\) for each \(k \le 8\) such that \(\mathrm{col_g}(G)=5.\) As a consequence, \(\mathrm{col_g}(\mathcal {P}_k)=5\) for \(k =7,8.\)

On the non-existence of antipodal cages of even girth

The Moore bound M(k,g) is a lower bound on the order of k-regular graphs of girth g (denoted (k,g)-graphs). The excess e of a (k,g)-graph of order n is the difference e=n−M(k,g). A (k,g)-cage is a (k,g)-graph with the fewest possible number of vertices. A graph of diameter d is said to be antipodal if, for any vertices u,v,w such that d(u,v)=d and d(u,w)=d, it follows that d(v,w)=d or v=w. Biggs and Ito proved that any (k,g)-cage of even girth g=2d≥6 and excess e≤k−2 is a bipartite graph of diameter d+1. In this paper we treat (k,g)-cages of even girth and excess e≤k−2. Based on spectral analysis we give a relation between the eigenvalues of the adjacency matrix A and the distance matrix Ad+1 of such cages. Applying matrix theory, we prove the non-existence of antipodal (k,g)-cages of excess e, for k≥e+2≥4 and g=2d≥14.

On the excess of vertex-transitive graphs of given degree and girth

We consider a restriction of the well-known Cage Problem to the class of vertex-transitive graphs, and consider the problem of finding the smallest vertex-transitive k-regular graphs of girth g. Counting cycles to obtain necessary arithmetic conditions on the parameters (k,g), we extend previous results of Biggs, and prove that, for any given excess e and any given degree k≥4, the asymptotic density of the set of girths g for which there exists a vertex-transitive (k,g)-cage with excess not exceeding e is 0.

Packing chromatic number of cubic graphs

A packingk-coloring of a graph G is a partition of V(G) into sets V1,…,Vk such that for each 1≤i≤k the distance between any two distinct x,y∈Vi is at least i+1. The packing chromatic number, χp(G), of a graph G is the minimum k such that G has a packing k-coloring. Sloper showed that there are 4-regular graphs with arbitrarily large packing chromatic number. The question whether the packing chromatic number of subcubic graphs is bounded appears in several papers. We answer this question in the negative. Moreover, we show that for every fixed k and g≥2k+2, almost every n-vertex cubic graph of girth at least g has the packing chromatic number greater than k.

Independent domination in subcubic graphs of girth at least six

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number, i(G), of G is the minimum cardinality of an independent dominating set. In this paper, we extend the work of Henning, Löwenstein, and Rautenbach (2014) who proved that if G is a bipartite, cubic graph of order n and of girth at least 6, then i(G)≤411n. We show that the bipartite condition can be relaxed, and prove that if G is a cubic graph of order n and of girth at least 6, then i(G)≤411n.

Infinitely many planar cubic hypohamiltonian graphs of girth 5

AbstractA graph G is hypohamiltonian if G is non‐hamiltonian and for every vertex v in G, the graph is hamiltonian. McKay asked in [J. Graph Theory 85 (2017) 7–11] whether infinitely many planar cubic hypohamiltonian graphs of girth 5 exist. We settle this question affirmatively.

Tight bounds on the coefficients of partition functions via stability

We show how to use the recently-developed occupancy method and stability results that follow easily from the method to obtain extremal bounds on the individual coefficients of the partition functions over various classes of bounded degree graphs.As applications, we prove new bounds on the number of independent sets and matchings of a given size in regular graphs. For large enough graphs and almost all sizes, the bounds are tight and confirm the Upper Matching Conjecture of Friedland, Krop, and Markström, and a conjecture of Kahn on independent sets for a wide range of parameters. Additionally we prove tight bounds on the number of q-colorings of cubic graphs with a given number of monochromatic edges, and tight bounds on the number of independent sets of a given size in cubic graphs of girth at least 5.

Convex and quasiconvex functions on trees and their applications

We introduce convex and quasiconvex functions on trees and prove that for a tree the eccentricity, transmission and weight functions are strictly quasiconvex. It is shown that the Perron vector of the distance matrix is strictly convex whereas the Perron vector of the distance signless Laplacian is quasiconvex for a tree. In the class of all trees with a given number of pendent vertices, we prove that the distance Laplacian and distance signless Laplacian spectral radius are both maximized at a dumbbell. Among all trees with fixed maximum degree, we prove that the broom is the unique tree that maximizes the distance Laplacian and distance signless Laplacian spectral radius. We find the unique graph that maximizes the distance spectral radius in the class of all unicyclic graphs of girth g on n vertices. Also we find the unique graph that maximizes the distance signless Laplacian and the distance Laplacian spectral radius in the class of all unicyclic graphs on n vertices.

On the difference between the Szeged and the Wiener index

We prove a conjecture of Nadjafi-Arani et al. on the difference between the Szeged and the Wiener index of a graph (Nadjafi-Aranifi et al., 2012). Namely, if G is a 2-connected non-complete graph on n vertices, then Sz(G)−W(G)≥2n−6. Furthermore, the equality is obtained if and only if G is the complete graph Kn−1 with an extra vertex attached to either 2 or n−2 vertices of Kn−1. We apply our method to strengthen some known results on the difference between the Szeged and the Wiener index of bipartite graphs, graphs of girth at least five, and the difference between the revised Szeged and the Wiener index. We also propose a stronger version of the aforementioned conjecture.

On the Minimum Edge‐Density of 4‐Critical Graphs of Girth Five

AbstractIn a recent seminal work, Kostochka and Yancey proved that for every 4‐critical graph G. In this article, we prove that for every 4‐critical graph G with girth at least five. When combined with another result of the second author, the improvement on the constant term leads to a corollary that there exist such that for every 4‐critical graph G with girth at least five. Moreover, it provides a unified and shorter proof of both a result of Thomassen and a result of Thomas and Walls without invoking any topological property, where the former proves that every graph with girth five embeddable in the projective plane or torus is 3‐colorable, and the latter proves the same for the Klein bottle.

Exhaustive generation of k‐critical ‐free graphs

AbstractWe describe an algorithm for generating all k‐critical ‐free graphs, based on a method of Hoàng et al. (A graph G is k‐critical H‐free if G is H‐free, k‐chromatic, and every H‐free proper subgraph of G is ‐colorable). Using this algorithm, we prove that there are only finitely many 4‐critical ‐free graphs, for both and . We also show that there are only finitely many 4‐critical ‐free graphs. For each of these cases we also give the complete lists of critical graphs and vertex‐critical graphs. These results generalize previous work by Hell and Huang, and yield certifying algorithms for the 3‐colorability problem in the respective classes. In addition, we prove a number of characterizations for 4‐critical H‐free graphs when H is disconnected. Moreover, we prove that for every t, the class of 4‐critical planar ‐free graphs is finite. We also determine all 52 4‐critical planar P7‐free graphs. We also prove that every P11‐free graph of girth at least five is 3‐colorable, and show that this is best possible by determining the smallest 4‐chromatic P12‐free graph of girth at least five. Moreover, we show that every P14‐free graph of girth at least six and every P17‐free graph of girth at least seven is 3‐colorable. This strengthens results of Golovach et al.