An odd coloring of a graph $G$ is a proper coloring of $G$ such that for every non-isolated vertex $v$, there is a color appearing an odd number of times in $N_G(v)$. Odd coloring of graphs was studied intensively in recent few years. In this paper, we introduce the notion of a strong odd coloring, as not only a strengthened version of odd coloring, but also a relaxation of square coloring. A strong odd coloring of a graph $G$ is a proper coloring of $G$ such that for every non-isolated vertex $v$, if a color appears in $N_G(v)$, then it appears an odd number of times in $N_G(v)$. We denote by $\chi_{so}(G)$ the smallest integer $k$ such that $G$ admits a strong odd coloring with $k$ colors. We prove that if $G$ is a graph with $mad(G)\le\frac{20}{7}$, then $\chi_{so}(G)\le \Delta(G)+4$, and the bound is tight. We also prove that if $G$ is a graph with $mad(G)\le\frac{30}{11}$ and $\Delta(G)\ge 4$, then $\chi_{so}(G)\le \Delta(G)+3$.

In 1969, Halin proved that every k-connected graph G with minimum degree at least k+1 contains an edge e such that G-e is k-connected. As an edge is a matching of size one, it is natural to ask whether Halin's result extends to matchings of larger size, a question recently investigated by Li, Zhou, Fujita, and Mao. A matching M of a k-connected graph G is called k-removable if G-M is k-connected. In this paper, we study minimum degree conditions that guarantee the existence of a k-removable matching of prescribed size. Specifically, we prove that for all positive integers k and m, every k-connected graph G with at least 2m vertices contains a k-removable matching of size m if δ(G) ≥ max{k+⌈ m2⌉, 2m} if k≥ m, k+m if k<m. As a consequence, every k-connected graph G with δ(G)≥2k+1 contains a k-removable matching of size ⌈(δ(G)+1)/2⌉, unless δ(G) is even and G≅ K_δ(G)+1. This verifies a conjecture of Li, Zhou, Fujita, and Mao in the range δ(G)≥2k+1. Our main tool, of independent interest, is a strengthening of Halin's result producing a k-removable edge that avoids a prescribed set of vertices.

Dean conjectured that for each integer k ≥ 3, every graph with minimum degree at least k has a cycle whose length is divisible by k; this conjecture is known to be true for all k≠ 5. For k∈{3,4}, stronger statements are true: every graph with minimum degree at least 2 and at most k-2 vertices of degree 2 has a cycle whose length is divisible by k. We further strengthen these results by characterizing all graphs with minimum degree at least 2 and at most three vertices of degree 2 that have no cycle of length divisible by k, for each k∈{3,4}. As a corollary, we obtain that every graph with minimum degree at least 2 and at most two vertices of degree 2 has a cycle whose length is divisible by 3, and that every graph on at least nine vertices with minimum degree at least 2 and at most three vertices of degree 2 has a cycle whose length is divisible by 4.

ABSTRACT For a graph , an ‐colouring of a graph is a vertex mapping such that adjacent vertices are mapped to adjacent vertices. A graph is ‐critical if has no ‐colouring but every proper subgraph of has a ‐colouring. We prove a structural characterisation of ‐critical series‐parallel graphs when . In the case that , we use the aforementioned characterisation to show a ‐free series‐parallel graph has a ‐colouring if either has neither nor , or has no two 5‐cycles sharing a vertex.

Mader conjectured that every k-strong digraph D with minimum semidegree δ^0(D)≥ 2k+m-1 contains a dipath P of order m such that D-V(P) remains k-strong. For k=1, he obtained the weaker bound δ^0(D)≥ 2m. We confirm the conjecture for k=1 by showing that the sharp bound δ^0(D)≥ m+1 suffices. As a consequence, we show that for every integer m≥2, every strongly connected digraph D with δ^0(D)≥max{2,m-1} contains a dipath P of order m such that D-A(P) is strongly connected.

For a graph G, a function f:V(G) →{0,1,2} is called a 2-limited dominating broadcast on G if for every vertex u, there exists a vertex v such that f(v)>0 and the distance between u and v in G is at most f(v). The cost of f means the value ∑_v∈ V(G)f(v), and the 2-limited broadcast domination number of G, denoted by γ_b,2(G), is the cost of a 2-limited dominating broadcast on G with minimum cost. Henning, MacGillivray, and Yang (2020) conjectured that γ_b,2(G)≤|V(G)|/3 for every cubic graph G. In this paper, we confirm the conjecture.

A longstanding conjecture of Seymour, called Seymour's second neighborhood conjecture, states that every oriented graph D contains a vertex x with |N^++_D(x)|≥ |N^+_D(x)|. The conjecture was verified in a few special classes of oriented graphs, and it remains open for general oriented graphs. We propose a stronger version of the conjecture that every oriented graph D contains a vertex x such that there exists a complete matching from N^+_D(x) to N^++_D(x). We prove that this stronger version holds for every oriented graph with minimum out-degree at most 5, and also for every 5-anti-transitive oriented graph. This implies that every oriented planar graph satisfies the stronger version.

Burr and Erdős conjectured in 1976 that for all integers $k>\ell\geq 0$ such that $k\mathbb{Z}+\ell$ contains an even integer, every $n$-vertex graph without cycles of length $\ell$ modulo $k$ has at most a linear number of edges in $n$. Bollobás confirmed the conjecture in 1977, and Erdős further asked for the exact extremal number. To the best of our knowledge, this problem has been solved only for all residues when $k\leq 4$, and for $\ell\in \{0,2\}$ when $k\geq 5$ is odd. In particular, Bai {\it et al.} [arXiv:2503.03504] proved that if $G$ is an $n$-vertex graph with no cycles of length $1$ modulo $3$, then $e(G)\le \frac{5}{3}(n-1)$, and when $9\mid (n-1)$ the equality holds if and only if each block of $G$ is isomorphic to the Petersen graph. Note that for $n> 18$ every extremal graph contains a cut-vertex. In this paper, we investigate the 2-connected setting and determine the maximum number of edges in a 2-connected graph with no cycles of length $1$ modulo $3$. Our results provide a sharp extremal bound and a complete characterization of the extremal graphs, revealing structural differences from the general case. Combining this with the result of Bai {\it et al.}, we also obtain a complete characterization of all extremal graphs in the general setting, including the cases where $9\nmid (n-1)$. Finally, we determine the maximum number of edges in a $2$-connected graph with no cycles of length $2$ modulo $4$, whose extremal graphs differ substantially from those in the general setting. Consequently, the extremal numbers for $2$-connected graphs with no cycle of a fixed length modulo $k$ are now determined for all $k\leq 4$.

In 2012, Mader conjectured that for any tree T of order m, every k-connected graph G with minimum degree at least ⌊3k/2⌋+m-1 contains a subtree T'≅ T such that G-V(T') remains k-connected. In 2022, Luo, Tian, and Wu considered an analogous problem for bipartite graphs and conjectured that for any tree T with bipartition (X,Y), every k-connected bipartite graph G with minimum degree at least k+max{|X|,|Y|} contains a subtree T'≅ T such that G-V(T') remains k-connected. In this paper, we relax the bipartite assumption by considering triangle-free graphs and prove that for any tree T of order m, every k-connected triangle-free graph G with minimum degree at least 2k+3m-4 contains a subtree T' ≅ T such that G-V(T') remains k-connected. Furthermore, we establish refined results for specific subclasses such as bipartite graphs or graphs with girth at least five.

An edge-coloured path is monochromatic if all of its edges have the same colour. For a k-connected graph G, the monochromatic k-connection number of G, denoted by mck(G), is the maximum number of colours in an edge-colouring of G such that, any two vertices are connected by k internally vertex-disjoint monochromatic paths. In this paper, we shall study the parameter mck(G). We obtain bounds for mck(G), for general graphs G. We also compute mck(G) exactly when k is small, and G is a graph on n vertices, with a spanning k-connected subgraph having the minimum possible number of edges, namely ⌈kn2⌉. We prove a similar result when G is a bipartite graph.

For a graph H, an H-colouring of a graph G is a vertex map ϕ:V(G) → V(H) such that adjacent vertices are mapped to adjacent vertices. A graph G is C_2k+1-critical if G has no C_2k+1-colouring but every proper subgraph of G has a C_2k+1-colouring. We prove a structural characterisation of C_2k+1-critical graphs when k ≥ 2. In the case that k = 2, we use the aforementioned charazterisation to show a C_3-free series-parallel graph G has a C_5-colouring if either G has neither C_8 nor C_10, or G has no two 5-cycles sharing a vertex.

The following relaxation of proper coloring the square of a graph was recently introduced: for a positive integer h, the proper h-conflict-free chromatic number of a graph G, denoted χpcfh(G), is the minimum k such that G has a proper k-coloring where every vertex v has min⁡{degG⁡(v),h} colors appearing exactly once on its neighborhood. Caro, Petruševski, and Škrekovski put forth a Brooks-type conjecture: if G is a graph with Δ(G)≥3, then χpcf1(G)≤Δ(G)+1. The best known result regarding the conjecture is χpcf1(G)≤2Δ(G)+1, which is implied by a result of Pach and Tardos. We improve upon the aforementioned result for all h, and also enlarge the class of graphs for which the conjecture is known to be true.Our main result is the following: for a graph G, if Δ(G)≥h+2, then χpcfh(G)≤(h+1)Δ(G)−1; this is tight up to the additive term as we explicitly construct infinitely many graphs G with χpcfh(G)=(h+1)(Δ(G)−1). We also show that the conjecture is true for chordal graphs, and obtain partial results for quasi-line graphs and claw-free graphs. Our main result also improves upon a Brooks-type result for h-dynamic coloring.

For two integers k and ℓ, an (ℓ mod k)-cycle means a cycle of length m such that m≡ℓk. In 1977, Bollobás proved a conjecture of Burr and Erdős by showing that if ℓ is even or k is odd, then every n-vertex graph containing no (ℓ mod k)-cycles has at most a linear number of edges in terms of n. Since then, determining the exact extremal bounds for graphs without (ℓ mod k)-cycles has emerged as an interesting question in extremal graph theory, though the exact values are known only for a few integers ℓ and k. Recently, Győri, Li, Salia, Tompkins, Varga and Zhu proved that every n-vertex graph containing no (0 mod 4)-cycles has at most ⌊19/12(n -1) ⌋ edges, and they provided extremal examples that reach the bound, all of which are not 2-connected. In this paper, we show that a 2-connected graph without (0 mod 4)-cycles has at most ⌊3n-1/2⌋ edges, and this bound is tight by presenting a method to construct infinitely many extremal examples.

A proper conflict-free c-coloring of a graph is a proper c-coloring such that each non-isolated vertex has a color appearing exactly once on its neighborhood. This notion was formally introduced by Fabrici et al., who proved that planar graphs have a proper conflict-free 8-coloring and constructed a planar graph with no proper conflict-free 5-coloring. Caro, Petruševski, and Škrekovski investigated this coloring concept further, and in particular studied upper bounds on the maximum average degree that guarantees a proper conflict-free c-coloring for c∈{4,5,6}.Along these lines, we completely determine the threshold on the maximum average degree of a graph G, denoted mad(G), that guarantees a proper conflict-free c-coloring for all c and also provide tightness examples. Namely, for c≥5 we prove that a graph G with mad(G)≤4cc+2 has a proper conflict-free c-coloring, unless G contains a 1-subdivision of the complete graph on c+1 vertices. When c=4, we show that a graph G with mad(G)<125 has a proper conflict-free 4-coloring, unless G contains an induced 5-cycle. In addition, we show that a planar graph with girth at least 5 has a proper conflict-free 7-coloring.

Assume G is a graph, (v_1,…,v_k) is a sequence of distinct vertices of G, and (a_1,…,a_k) is an integer sequence with a_i ∈{1,2}. We say G is (a_1,…,a_k)-list extendable (respectively, (a_1,…,a_k)-AT extendable) with respect to (v_1,…,v_k) if G is f-choosable (respectively, f-AT), where f(v_i)=a_i for i ∈{1,…, k}, and f(v)=3 for v ∈ V(G) ∖{v_1,…, v_k}. Hutchinson proved that if G is an outerplanar graph, then G is (2,2)-list extendable with respect to (x,y) for any vertices x,y. We strengthen this result and prove that if G is a K_4-minor-free graph, then G is (2,2)-AT extendable with respect to (x,y) for any vertices x,y. Then we characterize all triples (x,y,z) of a K_4-minor-free graph G for which G is (2,2,2)-AT extendable (as well as (2,2,2)-list extendable) with respect to (x,y,z). We also characterize the pairs (x,y) of a K_4-minor-free graph G for which G is (2,1)-AT extendable (as well as (2,1)-list extendable) with respect to (x,y). Moreover, we characterize all triples (x,y,z) of a 3-colorable graph G with its maximum average degree less than 14/5 for which G is (2,2,2)-AT extendable with respect to (x,y,z).

For a graph G, a subset S of V(G) is a hop dominating set of G if every vertex not in S has a 2-step neighbor in S. The hop domination number, γ_h(G), of G is the minimum cardinality of a hop dominating set of G. In this paper, we show that for a connected triangle-free graph G with n≥ 15 vertices, if δ(G)≥ 2, then γ_h(G)≤2n/5, and the bound is tight. We also give some tight upper bounds on γ_h(G) for triangle-free graphs G that contain a Hamiltonian path or a Hamiltonian cycle.

Given a graph $G$, a dominating set of $G$ is a set $S$ of vertices such that each vertex not in $S$ has a neighbor in $S$. The domination number of $G$, denoted $\gamma(G)$, is the minimum size of a dominating set of $G$. The independent domination number of $G$, denoted $i(G)$, is the minimum size of a dominating set of $G$ that is also independent. Recently, Abrishami and Henning proved that if $G$ is a cubic graph with girth at least $6$, then $i(G) \le \frac{4}{11}|V(G)|$. We show a result that not only improves upon the upper bound of the aforementioned result, but also applies to a larger class of graphs, and is also tight. Namely, we prove that if $G$ is a cubic graph without $4$-cycles, then $i(G) \le \frac{5}{14}|V(G)|$, which is tight. Our result also implies that every cubic graph $G$ without $4$-cycles satisfies $\frac{i(G)}{\gamma(G)} \le \frac{5}{4}$, which partially answers a question by O and West in the affirmative.

The domination number of a graph G $G$ , denoted γ ( G ) $\gamma (G)$ , is the minimum size of a dominating set of G $G$ , and the independent domination number of G $G$ , denoted i ( G ) $i(G)$ , is the minimum size of a dominating set of G $G$ that is also independent. Let k ≥ 4 $k\ge 4$ be an integer. Generalizing a result on cubic graphs by Lam, Shiu, and Sun, we prove that i ( G ) ≤ k − 1 2 k − 1 | V ( G ) | $i(G)\le \frac{k-1}{2k-1}|V(G)|$ for a connected k $k$ ‐regular graph G $G$ that is not K k , k ${K}_{k,k}$ , which is tight for k = 4 $k=4$ . This answers a question by Goddard et al. in the affirmative. We also show that i ( G ) γ ( G ) ≤ k 3 − 3 k 2 + 2 2 k 2 − 6 k + 2 $\frac{i(G)}{\gamma (G)}\le \frac{{k}^{3}-3{k}^{2}+2}{2{k}^{2}-6k+2}$ for a connected k $k$ ‐regular graph G $G$ that is not K k , k ${K}_{k,k}$ , strengthening upon a result of Knor, Škrekovski, and Tepeh. In addition, we prove that a graph G $G$ with maximum degree at most 4 satisfies i ( G ) ≤ 5 9 | V ( G ) | $i(G)\le \frac{5}{9}|V(G)|$ , which is also tight.

An odd c-coloring of a graph is a proper c -coloring such that each non-isolated vertex has a color appearing an odd number of times on its neighborhood. This concept was introduced very recently by Petruševski and Škrekovski and has attracted considerable attention. Cranston investigated odd colorings of graphs with bounded maximum average degree, and conjectured that every graph G with mad ( G ) ≤ 4 c − 4 c + 1 has an odd c -coloring for c ≥ 4, and proved the conjecture for c ∈ { 5 , 6 }. In particular, planar graphs with girth at least 7 and 6 have an odd 5-coloring and an odd 6-coloring, respectively. We completely resolve Cranston's conjecture. For c ≥ 7, we show that the conjecture is true, in a stronger form that was implicitly suggested by Cranston, but for c = 4, we construct counterexamples, which all contain 5-cycles. On the other hand, we show that a graph G with mad ( G ) < 22 9 and no induced 5-cycles has an odd 4-coloring. This implies that a planar graph with girth at least 11 has an odd 4-coloring. We also prove that a planar graph with girth at least 5 has an odd 6-coloring.

For a graph G, a function f:V(G)→{0,1,2} is called a 2-limited dominating broadcast on G if for every vertex u, there exists a vertex v such that f(v)>0 and the distance between u and v in G is at most f(v). The cost of f means the value ∑v∈V(G)f(v), and the 2-limited broadcast domination number γb,2(G) of G is the cost of a 2-limited dominating broadcast on G with minimum cost. Henning, MacGillivray and Yang (2020) conjectured that γb,2(G)≤|V(G)|3 for every cubic graph G, and then confirmed it for a cubic graph G having neither C4 nor C6 as an induced subgraph. In this paper, we improve their result, that is, we show that the conjecture holds for cubic graphs having no C4 as an induced subgraph.