A graph is called strongly _2k+1-connected if for each boundary function β: V(G)↦_2k+1 with ∑_v∈ V(G)β(v)≡ 02k+1, there exists an orientation D of G such that d_D^+(v) - d_D^-(v) ≡ β(v) 2k+1 for each v ∈ V(G). We show that every planar multigraph with 5 edge-disjoint spanning trees is strongly _5-connected. This verifies a special case of the Additive Base Conjecture when restricted to planar graphs. Hence, every 10-edge-connected directed planar graph admits an antisymmetric _5-flow. So, by duality, every orientation of a planar graph of girth at least 10 admits a homomorphism to a 5-vertex tournament. Our result also gives a new proof of the known result that every planar graph of girth at least 10 has a homomorphism to the 5-cycle.

Fix a positive integer r, and a graph G that is K-3,K- r-minor-free. Let I-s and I-t be two independent sets in G, each of size k. We begin with a "token" on each vertex of I-s and seek to move all tokens to I-t, by repeated "token jumping", removing a single token from one vertex and placing it on another vertex. We require that each intermediate arrangement of tokens again specifies an independent set of size k. Given G, I-s, and I-t, we ask whether there exists a sequence of token jumps that transforms I-s into I-t. When k is part of the input, this problem is known to be PSPACE-complete. But it was shown by Ito, Kaminski, and Ono [20] to be fixed-parameter tractable. That is, the problem can be solved in time f(k) center dot P(n), for some function f and polynomial P, where n denotes the order of G. Here we strengthen the upper bound on the running time in terms of k by showing that the problem has a kernel of size linear in k. More precisely, we transform an arbitrary input problem on a K-3,K- r-minor-free graph (for some fixed positive integer r) into an equivalent problem on a (K(3, r-)minor-free) graph with order O(k). This answers positively a question of Bousquet, Mouawad, Nishimura, and Siebertz [13] and improves the recent quadratic kernel of Cranston, Muhlenthaler, and Peyrille [14]. For planar graphs, we further strengthen this upper bound to get a kernel of size at most 42k.

ABSTRACT A graph is called strongly ‐connected if for each boundary function with , there exists an orientation of such that for each . We show that every planar multigraph with 5 edge‐disjoint spanning trees is strongly ‐connected. This verifies a special case of the Additive Base Conjecture when restricted to planar graphs. Hence, every 10‐edge‐connected directed planar graph admits an antisymmetric ‐flow. By duality, every orientation of a planar graph of girth at least 10 admits a homomorphism to a 5‐vertex tournament. Our result also gives a new proof of the known result that every planar graph of girth at least 10 has a homomorphism to the 5‐cycle.

Fix an abelian group A, a graph G, and nowhere-zero A-flows f' and f” on G. Now f' and f” are A-flow-adjacent if there exists a cycle C in G such that f'(e)-f”(e)=0 for all edges e∉ E(C). And f' and f” are A-flow-equivalent if there exists a sequence f_0,…,f_s of A-flows such that f_0=f', f_s=f”, and f_i and f_i-1 are A-flow-adjacent for all i∈[s]. Given a group A, we seek conditions on a graph G such that all A-flows on G are pairwise A-flow-equivalent; in this case, we say that G is A-flow-connected. Analogously, we define k-flow-connectedness for nowhere-zero (integer) k-flows. The notions of A-flow-connectedness and k-flow-connectedness were first investigated by Esperet et al., who showed, among other results, that every 2-edge-connected graph is A-flow-connected whenever A=ℤ_2^8 or |A| ≥ 1.15× 10^694. In this paper, we first characterize the graphs that are ℤ_3-flow-connected and that are 3-flow-connected. We show that every 2-edge-connected graph is A-flow-connected if and only if this is true for every 2-edge-connected cubic graphs. We show that all cubic bipartite graphs are ℤ_4-flow-connected, and construct other cubic graphs that are and are not ℤ_4-flow-connected. We conjecture that every Eulerian graph is k-flow-connected and A-flow-connected whenever k or |A| is even; and provide evidence for this conjecture. Finally, we consider 4-edge-connected graphs G. Here, we show that G is A-flow-connected whenever |A|≥ 5.3× 10^6.

The problem Token Jumping asks whether, given a graph G and two independent sets of tokens I and J of G, we can transform I into J by changing the position of a single token in each step and having an independent set of tokens throughout. We show that there is a polynomial-time algorithm that, given an instance of Token Jumping, computes an equivalent instance of size O((g + k)^2) , where g is the genus of the input graph and k is the size of the independent sets. Our algorithm is very simple and does not require any information about the genus of the input graph.

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.

Albertson conjectured that every graph with chromatic number r has crossing number at least the crossing number of the complete graph K_r. This conjecture was proved for r≤ 12 by Albertson, Cranston, and Fox; for r≤ 16 by Barát and Tóth; and for r≤ 18 by Ackerman. Here we verify it for r≤ 24; we also greatly restrict the possibilities for counterexamples when r∈{25,26}. In addition, we strengthen earlier work bounding the order of a minimum counterexample for each choice of r: we exclude the possibility that |G|≥ 2.82r and exclude the possibility that 1.228r≤ |G|≤ 1.768r. Finally, as r grows, we extend the lower end of this range of excluded orders for a minimum counterexample. In particular: if r≥ 125,000, then we exclude the possibility that 1.10r≤ |G|≤ 1.768r; and if r≥ 825,000, then we exclude the possibility that 1.05r≤ |G|≤ 1.768r.

Fix a planar graph G and a list-assignment L with |L(v)|=10 for all v∈ V(G). Let α and β be L-colorings of G. A recoloring sequence from α to β is a sequence of L-colorings, beginning with α and ending with β, such that each successive pair in the sequence differs in the color on a single vertex of G. We show that there exists a constant C such that for all choices of α and β there exists a recoloring sequence σ from α to β that recolors each vertex at most C times. In particular, σ has length at most C|V(G)|. This confirms a conjecture of Dvořák and Feghali. For our proof, we introduce a new technique for quickly showing that many configurations are reducible. We believe this method may be of independent interest and will have application to other problems in this area.

For a graph G and a list assignment L with IL(v)I = k for all v, an L-packing consists of L-colorings phi(1), ..., phi(k) such that phi i(v) not equal phi j(v) for all vand all distinct i, jE {1, ...,k}. Let chi(star) (G) denote the smallest k such that G has an L-packing for every L with IL(v)I = kfor all v. Let P-k denote the set of all planar graphs with girth at least k. We show that (i) chi e (G) <= 8 for all G E P-3 and (ii) chi (G) 5 <= for all G E P-4 and (iii) chi (G) 4 <= for all G E P-5. Part (i) makes progress on a problem of Cambie, Cames van Batenburg, Davies, and Kang. We also consider the analogue of chi e for correspondence coloring, chi c star. In fact, all bounds stated above for chi e also hold for chi c .

A proper s-coloring of an n-vertex graph is equitable if every color class has size ⌊n/s⌋ or ⌈n/s⌉. A necessary condition to have an equitable s-coloring is that every vertex v appears in an independent set of size at least ⌊n/s⌋. That is min_v∈ V(G)α_v≥⌊n/s⌋. Various authors showed that when G is a tree and s≥ 3 this obvious necessary condition is also sufficient. Kierstead, Kostochka, and Xiang asked whether this result holds more generally for all outerplanar graphs. We show that the answer is No when s=3, but that the answer is Yes when s≥ 6. The case s∈{4,5} remains open. We also prove an analogous result for planar graphs, with a necessary and sufficient hypothesis. Fix s≥ 40. Let G be a planar graph, and let w_0,w_1 be its 2 vertices with largest degrees. If there exist disjoint independent sets I_0, I_1 such that |I_0|=⌊n/s⌋ and |I_1| = ⌊(n+1)/s⌋ and w_0,w_1∈ I_0∪ I_1, then G has an equitable s-coloring.

For every r≥13, we show every 1-planar graph G with Δ(G)≤r has an equitable r-coloring.

Let $G$ be a planar graph and $I_s$ and $I_t$ be two independent sets in $G$, each of size $k$. We begin with a ``token'' on each vertex of $I_s$ and seek to move all tokens to $I_t$, by repeated ``token jumping'', removing a single token from one vertex and placing it on another vertex. We require that each intermediate arrangement of tokens again specifies an independent set of size $k$. Given $G$, $I_s$, and $I_t$, we ask whether there exists a sequence of token jumps that transforms $I_s$ to $I_t$. When $k$ is part of the input, this problem is known to be PSPACE-complete. However, it was shown by Ito, Kami\'nski, and Ono to be fixed-parameter tractable. That is, when $k$ is fixed, the problem can be solved in time polynomial in the order of $G$. Here we strengthen the upper bound on the running time in terms of $k$ by showing that the problem has a kernel of size linear in $k$. More precisely, we transform an arbitrary input problem on a planar graph into an equivalent problem on a (planar) graph with order $O(k)$.

Let $G$ be a graph and $k$ be a positive integer, and let $Kc(G, k)$ denote the number of Kempe equivalence classes for the $k$-colorings of $G$. In 2006, Mohar noted that $Kc(G, k) = 1$ if $G$ is bipartite. As a generalization, we show that $Kc(G, k) = 1$ if $G$ is formed from a bipartite graph by adding any number of edges less than $\binom{\lceil k/2\rceil}2+\binom{\lfloor k/2\rfloor}2$. We show that our result is tight (up to lower order terms) by constructing, for each $k \geq 8$, a graph $G$ formed from a bipartite graph by adding $(k^2+8k-45+1)/4$ edges such that $Kc(G, k) \geq 2$. This refutes a recent conjecture of Higashitani--Matsumoto.

Hocquard, Kim, and Pierron constructed, for every even integer , a 2‐degenerate graph with maximum degree such that . We prove for (a) all 2‐degenerate graphs and (b) all graphs with , upper bounds on the clique number of that match the lower bound given by this construction, up to small additive constants. We show that if is 2‐degenerate with maximum degree , then (with when is sufficiently large). And if has and maximum degree , then . Thus, the construction of Hocquard et al. is essentially the best possible. Our proofs introduce a “token passing” technique to derive crucial information about nonadjacencies in of vertices that are adjacent in . This is a powerful technique for working with such graphs that has not previously appeared in the literature.

A coloring of a graph G is a map f : V(G)-> Z(+) such that f (v)not equal f (w) for all vw is an element of E(G). A coloring f is an odd-sum coloring if Sigma(w is an element of N[v]) f (w) is odd, for each vertex v is an element of V (G). The odd-sum chromatic number of a graph G, denoted chi(os)(G), is the minimum number of colors used (that is, the minimum size of the range) in an odd-sum coloring of G. Caro, Petrusevski, and Skrekovski showed, among other results, that chi(os)(G) is well-defined for every finite graph G and , in fact, chi(os)(G) <= 2 chi(G). Thus, chi(os)(G) <= 8 for every planar graph G (by the 4 Color Theorem), chi(os)(G) <= 6 for every triangle-free planar graph G (by Grotzsch's Theorem), and chi(os)(G) <= 4 for every bipartite graph. Caro et al. asked, for every even Delta >= 4, whether there exists g(Delta) such that if G is planar with maximum degree Delta and girth at least g increment then chi(os)(G) <= 5. They also asked, for every even Delta >= 4, whether there exists g(Delta) such that if G is planar and bipartite with maximum degree Delta and girth at least g increment then chi(os)(G) <= 3. We answer both questions negatively. We also refute a conjecture they made, resolve one further problem they posed, and make progress on another. (c) 2023 Elsevier B.V. All rights reserved.

The coloring reconfiguration graph Ck(G) ${{\mathscr{C}}}_{k}(G)$ has as its vertex set all the proper k $k$-colorings of G $G$, and two vertices in Ck(G) ${{\mathscr{C}}}_{k}(G)$ are adjacent if their corresponding k $k$-colorings differ on a single vertex. Cereceda conjectured that if an n $n$-vertex graph G $G$ is d $d$-degenerate and k >= d+2 $k\ge d+2$, then the diameter of Ck(G) ${{\mathscr{C}}}_{k}(G)$ is O(n2) $O({n}<^>{2})$. Bousquet and Heinrich proved that if G $G$ is planar and bipartite, then the diameter of C5(G) ${{\mathscr{C}}}_{5}(G)$ is O(n2) $O({n}<^>{2})$. (This proves Cereceda's Conjecture for every such graph with degeneracy 3.) They also highlighted the particular case of Cereceda's Conjecture when G $G$ is planar and has no 3-cycles. As a partial solution to this problem, we show that the diameter of C5(G) ${{\mathscr{C}}}_{5}(G)$ is O(n2) $O({n}<^>{2})$ for every planar graph G $G$ with no 3-cycles and no 5-cycles.

A proper coloring of a graph is called \emph{odd} if every non-isolated vertex has some color that appears an odd number of times on its neighborhood. The smallest number of colors that admits an odd coloring of a graph $G$ is denoted $\chi_o(G)$. This notion was introduced by Petru\v{s}evski and \v{S}krekovski, who proved that if $G$ is planar then $\chi_o(G)\le 9$; they also conjectured that $\chi_o(G)\le 5$. For a positive real number $\alpha$, we consider the maximum value of $\chi_o(G)$ over all graphs $G$ with maximum average degree less than $\alpha$; we denote this value by $\chi_o(\mathcal{G}_{\alpha})$. We note that $\chi_o(\mathcal{G}_{\alpha})$ is undefined for all $\alpha\ge 4$. In contrast, for each $\alpha\in[0,4)$, we give a (nearly sharp) upper bound on $\chi_o(\mathcal{G}_{\alpha})$. Finally, we prove $\chi_o(\mathcal{G}_{20/7})= 5$ and $\chi_o(\mathcal{G}_3)= 6$. Both of these results are sharp.

A proper coloring of a graph is conflict-free if, for every nonisolated vertex, some color is used exactly once on its neighborhood. Caro, Petruvsevski, and Skrekovski [Discrete Math., 346 (2023), 113221] proved that every graph G has a proper conflict-free coloring with at most Delta(G)/2 colors and conjectured that \Delta (G) + 1 colors suffice for every connected graph G with Delta(G) >= 3. Our first main result is that even for list-coloring, inverted right perpendicular 1.6550826 Delta (G)+ root Delta (G)inverted left perpendicular colors suffice for every graph G with Delta (G) >= 108; we also prove slightly weaker bounds for all graphs with Delta(G) >= 750. These results follow from our more general framework on proper conflict-free list-coloring of a pair consisting of a graph G and a "conflict" hypergraph H. As another corollary of our results in this general framework, every graph has a proper (root 30 + o(1))Delta(G)1.5-list-coloring such that every bichromatic component is a path on at most three vertices, where the number of colors is optimal up to a constant factor. Our proof uses a fairly new type of recursive counting argument called Rosenfeld counting, which is a variant of the Lovasz local lemma or entropy compression. We also prove an asymptotically optimal result for a fractional analogue of our general framework for proper conflict-free coloring for pairs of a graph and a conflict hypergraph. A corollary states that every graph G has a fractional (1+ o(1))Delta(G)-coloring such that every fractionally bichromatic component has at most two vertices. In particular, it implies that the fractional analogue of the conjecture of Caro, Petruvsevski, and Skrekovski holds asymptotically in a strong sense.

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.

Wegner conjectured that if G is a planar graph with maximum degree Δ≥8, then χ(G2)≤32Δ+1. This problem has received much attention, but remains open for all Δ≥8. Here we prove an analogous bound on ω(G2): If G is a plane graph with Δ(G)≥36, then ω(G2)≤⌊32Δ(G)⌋+1. In fact, this is a corollary of the following lemma, which is our main result. If G is a plane graph with Δ(G)≥19 and S is a maximal clique in G2 with |S|≥Δ(G)+20, then there exist x,y,z∈V(G) such that S={w:|N[w]∩{x,y,z}|≥2}.