We show that the k-colour Ramsey number of an odd cycle of length 2 & ell; + 1 is at most (4 & ell;)k & centerdot; kk/& ell;. This proves a conjecture of Fox and is the first improvement in the exponent that goes beyond an absolute constant factor since the work of Bondy and Erd & odblac;s from 1973. (c) 2026 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

In 1979, Albertson and Berman conjectured that every planar graph G contains an induced forest of order at least |V(G)|/2. This long-standing conjecture was recently disproved by several explicit counterexamples, which naturally led to several extremal and structural questions that we answer. We combine mathematical arguments and exhaustive computations to show that the minimum order of a counterexample is 29. We also construct infinitely many 4-connected 5-edge-connected counterexamples (and show that the unique such counterexample of minimum order has order 41), whereas previously all known counterexamples had vertex-connectivity at most 3. Furthermore, we construct an infinite family of planar graphs on n vertices whose maximum induced forests have order at most 25/52n, thereby improving the previous best upper bound. This family also yields infinitely many counterexamples (for every integer d ≥ 7) to a conjecture of Chappell and Pelsmajer concerning induced forests of maximum degree at most d.

The domination number γ(G) of a graph G is the smallest possible size of a vertex set that intersects every radius-1 ball of G, and the packing number ρ(G) is the maximum number of pairwise vertex-disjoint radius-1 balls. We prove that γ(G)/ρ(G)≤ 5 for every planar graph and γ(G)/ρ(G)≤18√(3)/π≈ 9.924 for every unit disk graph, thus yielding Erdős-Pósa-type bounds for the hypergraph of radius-1 balls in the two graph classes. This improves upon results of Gutiérrez and Paul, and Dúcz and Gujgiczer, who in turn lowered bounds of Bonamy, Csikós, Gujgiczer and Yuditsky, and Böhme and Mohar. For both graph classes, the best known lower bound on the optimal constant remains 3.

The fractional list packing number of a graph is a graph invariant that has recently arisen from the study of disjoint list-colourings. It measures how large the lists of a list-assignment need to be to ensure the existence of a "perfectly balanced" probability distribution on proper -colourings, that is, such that at every vertex , every colour appears with equal probability . In this work we give various bounds on , which admit strengthenings for correspondence and local-degree versions. As a corollary, we improve theorems on the related notion of flexible list colouring. In particular we study Cartesian products and -degenerate graphs, and we prove that is bounded from above by the pathwidth of plus one. The correspondence analogue of the latter is false for treewidth instead of pathwidth.

A recolouring sequence, between k-colourings alpha and beta of a graph G, transforms alpha into beta by recolouring one vertex at a time, such that after each recolouring step we again have a proper k-colouring of G. The diameter of the k-recolouring graph, diam Ck(G), is the maximum over all pairs alpha and beta of the minimum length of a recolouring sequence from alpha to beta. Much previous work has focused on determining the asymptotics of diam Ck(G): Is it Theta(|G|)? Is it Theta(|G|2)? Or even larger? Here we focus on graphs for which diam Ck(G) = Theta(|G|), and seek to determine more precisely the multiplicative constant implicit in the Theta(). In particular, for each k 3, for all positive integers p and q we exactly determine diam Ck(Kp,q), up to a small additive constant. We also sharpen a recolouring lemma that has been used in multiple papers, proving an optimal version. This improves the multiplicative constant in various prior results. Finally, we investigate plausible relationships between similar reconfiguration graphs.

Thomassen famously proved that every planar graph is 5-choosable. We explore variants of this result, focusing on finding disjoint correspondence colorings, in the more general class of K_5-minor-free graphs. Correspondence colorings generalize list colorings as follows. Given a graph G and a positive integer t, a correspondence t-cover M assigns to each v∈ V(G) a set of allowable colors {1_v,…,t_v} and to each edge vw∈ E(G) a matching between {1_v,…,t_v} and {1_w,…,t_w}. An M-coloring φ picks for each vertex v a color φ(v) (from the set {1_v,…,t_v}) such that for each edge vw∈ E(G) the colors φ(v),φ(w) are not matched to each other. Two M-colorings φ_1,φ_2 of G are called disjoint if φ_1(v)_2(v) for all v∈ V(G). For every K_5-minor-free graph G and every correspondence 6-cover M of G, we construct 3 pairwise disjoint M-colorings φ_1,φ_2,φ_3. In contrast, we provide examples of K_5-minor-free graphs and correspondence 5-covers M that do not admit 3 disjoint M-colorings.

A graph class C has polynomial expansion if there is a polynomial function f such that for every graph G is an element of C, each of the depth-r minors of G has average degree at most f (r). In this note, we study bounded-radius variants of some classical graph parameters such as bramble number, linkedness and well-linkedness, and we show that they are pairwise polynomially related. Furthermore, in a graph class with polynomial expansion they are all uniformly bounded by a polynomial in r.

Given a proper (list) colouring of a graph G, a recolouring step changes the colour at a single vertex to another colour (in its list) that is currently unused on its neighbours, hence maintaining a proper colouring. Suppose that each vertex v has its own private list L(v) of allowed colours such that |L(v)|≥(v)+1. We prove that if G is connected and its maximum degree Δ is at least 3, then for any two proper L-colourings in which at least one vertex can be recoloured, one can be transformed to the other by a sequence of O(|V(G)|^2) recolouring steps. We also show that reducing the list-size of a single vertex w to (w) can lead to situations where the space of proper L-colourings is `shattered'. Our results can be interpreted as showing a sharp phase transition in the Glauber dynamics of proper L-colourings of graphs. This constitutes a `local' strengthening and generalisation of a result of Feghali, Johnson, and Paulusma, which considered the situation where the lists are all identical to {1,…,Δ+1}.

We investigate the list packing number of a graph, the least $k$ such that there are always $k$ disjoint proper list-colourings whenever we have lists all of size $k$ associated to the vertices. We are curious how the behaviour of the list packing number contrasts with that of the list chromatic number, particularly in the context of bounded degree graphs. The main question we pursue is whether every graph with maximum degree $\Delta$ has list packing number at most $\Delta+1$. Our results highlight the subtleties of list packing and the barriers to, for example, pursuing a Brooks'-type theorem for the list packing number.

List colouring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list-colouring, we seek many in parallel. Our explorations have uncovered a potentially rich seam of interesting problems spanning chromatic graph theory. Given a k-list-assignment L of a graph G, which is the assignment of a list L(v) of k colours to each vertex v∈ V(G), we study the existence of k pairwise-disjoint proper colourings of G using colours from these lists. We may refer to this as a list-packing. Using a mix of combinatorial and probabilistic methods, we set out some basic upper bounds on the smallest k for which such a list-packing is always guaranteed, in terms of the number of vertices, the degeneracy, the maximum degree, or the (list) chromatic number of G. (The reader might already find it interesting that such a minimal k is well defined.) We also pursue a more focused study of the case when G is a bipartite graph. Our results do not yet rule out the tantalising prospect that the minimal k above is not too much larger than the list chromatic number. Our study has taken inspiration from study of the strong chromatic number, and we also explore generalisations of the problem above in the same spirit.

The reconfiguration graph Ck(G) for the k-colourings of a graph G has a vertex for each proper k-colouring of G, and two vertices of Ck(G) are adjacent precisely when those k-colourings differ on a single vertex of G. Much work has focused on bounding the maximum value of diamCk(G) over all n-vertex graphs G. We consider the analogous problems for list colourings and for correspondence colourings. We conjecture that if L is a list-assignment for a graph G with |L(v)|≥d(v)+2 for all v∈V(G), then diamCL(G)≤n(G)+μ(G). We also conjecture that if (L,H) is a correspondence cover for a graph G with |L(v)|≥d(v)+2 for all v∈V(G), then diamC(L,H)(G)≤n(G)+τ(G). (Here μ(G) and τ(G) denote the matching number and vertex cover number of G.) For every graph G, we give constructions showing that both conjectures are best possible, which also hints towards an exact form of Cereceda’s Conjecture for regular graphs. Our first main result proves the upper bounds (for the list and correspondence versions, respectively) diamCL(G)≤n(G)+2μ(G) and diamC(L,H)(G)≤n(G)+2τ(G). Our second main result proves that both conjectured bounds hold, whenever all v satisfy |L(v)|≥2d(v)+1. We conclude by proving one or both conjectures for various classes of graphs such as complete bipartite graphs, subcubic graphs, cactuses, and graphs with bounded maximum average degree. The full paper can also be found at arxiv.org/abs/2204.07928.

One of Thomassen's classical results is that every planar graph of girth at least 5 is 3-choosable. One can wonder if for a planar graph G of girth sufficiently large and a 3-list-assignment L, one can do even better. Can one find 3 disjoint L-colorings (a packing), or 2 disjoint L-colorings, or a collection of L-colorings that to every vertex assigns every color on average in one third of the cases (a fractional packing)? We prove that the packing is impossible, but two disjoint L-colorings are guaranteed if the girth is at least 8, and a fractional packing exists when the girth is at least 6. For a graph G, the least k such that there are always k disjoint proper list-colorings whenever we have lists all of size k associated to the vertices is called the list packing number of G. We lower the two-times-degeneracy upper bound for the list packing number of planar graphs of girth 3,4 or 5. As immediate corollaries, we improve bounds for ϵ-flexibility of classes of planar graphs with a given girth. For instance, where previously Dvořák et al. proved that planar graphs of girth 6 are (weighted) ϵ-flexibly 3-choosable for an extremely small value of ϵ, we obtain the optimal value ϵ=1/3. Finally, we completely determine and show interesting behavior on the packing numbers for H-minor-free graphs for some small graphs H.

We prove that every connected cubic graph with $n$ vertices has a maximal matching of size at most $\frac{5}{12} n+ \frac{1}{2}$. This confirms the cubic case of a conjecture of Baste, Fürst, Henning, Mohr and Rautenbach (2019) on regular graphs. More generally, we prove that every graph with $n$ vertices and $m$ edges and maximum degree at most $3$ has a maximal matching of size at most $\frac{4n-m}{6}+ \frac{1}{2}$. These bounds are attained by the graph $K_{3,3}$, but asymptotically there may still be some room for improvement. Moreover, the claimed maximal matchings can be found efficiently. As a corollary, we have a $\left(\frac{25}{18} + O \left( \frac{1}{n}\right)\right) $-approximation algorithm for minimum maximal matching in connected cubic graphs.

We wish to bring attention to a natural but slightly hidden problem, posed by Erdös and Nešetřil in the late 1980s, an edge version of the degree--diameter problem. Our main result is that, for any graph of maximum degree $\Delta$ with more than $1.5 \Delta^t$ edges, its line graph must have diameter larger than $t$. In the case where the graph contains no cycle of length $2t+1$, we can improve the bound on the number of edges to one that is exact for $t\in\{1,2,3,4,6\}$. In the case $\Delta=3$ and $t=3$, we obtain an exact bound. Our results also have implications for the related problem of bounding the distance-$t$ chromatic index, $t>2$; in particular, for this, we obtain an upper bound of $1.941\Delta^t$ for graphs of large enough maximum degree $\Delta$, markedly improving on earlier bounds for this parameter.

We investigate the effect of a fixed forbidden clique minor upon the strong chromatic index, both in multigraphs and in simple graphs. We conjecture for each k >= 4 that any K k-minor-free multigraph of maximum degree Delta has strong chromatic index at most 3 2 ( k - 2 ) Delta. We present a construction certifying that if true the conjecture is asymptotically sharp as Delta -> infinity. In support of the conjecture, we show it in the case k = 4 and prove the statement for strong clique number in place of strong chromatic index. By contrast, we make a basic observation that for K k-minor-free simple graphs, the problem of strong edge-colouring is "between" Hadwiger's Conjecture and its fractional relaxation. For k >= 5, we also show that K k-minor-free multigraphs of edge-diameter at most 2 have strong clique number at most ( k - 1 2 ) Delta.

Motivated by a recent conjecture of the first author, we prove that every properly coloured triangle-free graph of chromatic number $\chi$ contains a rainbow independent set of size $\lceil\frac12\chi\rceil$. This is sharp up to a factor $2$. This result and its short proof have implications for the related notion of chromatic discrepancy. Drawing inspiration from both structural and extremal graph theory, we conjecture that every triangle-free graph of chromatic number $\chi$ contains an induced cycle of length $\Omega(\chi\log\chi)$ as $\chi\to\infty$. Even if one only demands an induced path of length $\Omega(\chi\log\chi)$, the conclusion would be sharp up to a constant multiple. We prove it for regular girth $5$ graphs and for girth $21$ graphs. As a common strengthening of the induced paths form of this conjecture and of Johansson's theorem (1996), we posit the existence of some $c >0$ such that for every forest $H$ on $D$ vertices, every triangle-free and induced $H$-free graph has chromatic number at most $c D/\log D$. We prove this assertion with 'triangle-free' replaced by 'regular girth 5'.

We prove that for every integer t ⩾ 1 there exists a constant ct such that for every Kt-minor-free graph G, and every set S of balls in G, the minimum size of a set of vertices of G intersecting all the balls of S is at most ct times the maximum number of vertex-disjoint balls in S. This was conjectured by Chepoi, Estellon, and Vaxès in 2007 in the special case of planar graphs and of balls having the same radius.

Given a graph G, the strong clique number ω2′(G) of G is the cardinality of a largest collection of edges every pair of which are incident or connected by an edge in G. We study the strong clique number of graphs missing some set of cycle lengths. For a graph G of large enough maximum degree Δ, we show among other results the following: ω2′(G)≤5Δ2∕4 if G is triangle-free; ω2′(G)≤3(Δ−1) if G is C4-free; ω2′(G)≤Δ2 if G is C2k+1-free for some k≥2. These bounds are attained by natural extremal examples. Our work extends and improves upon previous work of Faudree, Gyárfás, Schelp and Tuza (1990), Mahdian (2000) and Faron and Postle (2019). We are motivated by the corresponding problems for the strong chromatic index.

We prove that any triangle-free graph on $n$ vertices with minimum degree at least $d$ contains a bipartite induced subgraph of minimum degree at least $d^2/(2n)$. This is sharp up to a logarithmic factor in $n$. Relatedly, we show that the fractional chromatic number of any such triangle-free graph is at most the minimum of $n/d$ and $(2+o(1))\sqrt{n/\log n}$ as $n\to\infty$. This is sharp up to constant factors. Similarly, we show that the list chromatic number of any such triangle-free graph is at most $O(\min\{\sqrt{n},(n\log n)/d\})$ as $n\to\infty$. Relatedly, we also make two conjectures. First, any triangle-free graph on $n$ vertices has fractional chromatic number at most $(\sqrt{2}+o(1))\sqrt{n/\log n}$ as $n\to\infty$. Second, any triangle-free graph on $n$ vertices has list chromatic number at most $O(\sqrt{n/\log n})$ as $n\to\infty$.

A classic theorem of Erdos and Posa [Canal. T. Math., 17 (1965), pp. 347-352] states that every graph has either k vertex-disjoint cycles or a set of O(k log k) vertices meeting all its cycles. While the standard proof revolves around finding a large "frame" in the graph (a subdivision of a large cubic graph), an alternative way of proving this theorem is to use a ball packing argument of Kuhn and Osthus [Random Structures Algorithms, 22 (2003), pp. 213-225] and Diestel and Rempel [Combinatorica, 25 (2005), pp. 111-116]. In this paper, we argue that the latter approach is particularly well suited for studying edge variants of the Erdos-Posa theorem. As an illustration, we give a short proof of a theorem of Bruhn, Heinlein, and Joos [Combinatorica, 39 (2019), pp. 1-36] that cycles of length at least l have the so-called edge-Erdos-Posa property. More precisely, we show that every graph G contains either k edge-disjoint cycles of length at least l or an edge set F of size O(kl . log(kl)) such that G - F has no cycle of length at least l. For fixed l, this improves on the previously best known bound of O(k(2) log k + kl).