ABSTRACT A graph is called ‐degenerate if every subgraph of contains a vertex of degree at most . It is known that planar graphs are 5‐degenerate and that every planar graph without ‐cycles for some prescribed is 3‐degenerate. In this paper, we prove that for every prescribed integer , every planar graph without ‐cycles is 4‐degenerate. This result is the best possible in the sense that for each there exists a planar graph without ‐cycles and .
ABSTRACT Let be the smallest integer such that every bridgeless graph with has a strong orientation satisfying . In 1978, Chvátal and Thomassen proved . Later, Babu–Benson–Rajendraprasad–Vaka improved the upper bound to . We show that the leading constant can be reduced to 1: We also prove the explicit bound for every .
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.
ABSTRACT The line graph of a multigraph is the graph whose vertices are the edges of , where two such edges are adjacent if and only if they meet in a single vertex of . We provide several characterizations of such line graphs and in particular show that a graph is a line graph if and only if it does not contain one of the 32 graphs, all of which correspond to bases of nonisotropic vectors of a six‐dimensional quadratic geometry of −‐type over a field with two elements, or, equivalently, to sets of six generating reflections in the Weyl group of type . Beineke's well‐known characterization of line graphs of ordinary graphs by nine forbidden subgraphs is a special case of our result, just as the characterization of generalized line graphs by 31 forbidden subgraphs by Cvetković, Doob, and Simić.
ABSTRACT For a graph and a graph family , let denote the maximum number of copies of in an ‐free ‐vertex graph. Let . Bai, Tompkins, and Well conjectured that is attained if and each block of the graph is a . In this paper, we determine the exact value of and the extremal graphs for all . The novelty of our proof is to give a proper partition of the set of triangles in an extremal graph. On the basis of this partition, we obtain the partition of the edge set and thus the structure of an extremal graph. Our new method can also be applied to obtain some meaningful results in other settings.
ABSTRACT A hole is an induced cycle of length at least 4, an even hole is a hole of even length, and a cap is a graph consisting of a hole and an additional vertex which has exactly two neighbors in the hole that are adjacent in the hole. A graph obtained from a graph by blowing up all the vertices into cliques is said to be a clique blowup of . In this paper, we introduce a new method and transfer the optimal binding function problem for the class of (cap, even‐hole)‐free graphs to those with clique number at most 4. Specificly, we show that every (cap, even hole)‐free graph satisfies , which affirmatively answers a question of Cameron et al. [19], we also show that every (cap, even hole, )‐free graph satisfies . Both bounds are tight.
ABSTRACT Let be a connected graph. If is a set of vertices of , and is a vertex of , then the distance is the distance between and a vertex of closest to . The eccentricity of is defined as . For , the ‐radius of is defined as the smallest eccentricity of a set of vertices of . The ‐radius is, in some sense, a dual to the ‐distance domination number of , defined as the minimum cardinality of a set of vertices of such that every vertex not in is within distance of some vertex in . We prove that the ‐radius of a 2‐connected graph of order is at most . We also extend the well‐known result that the vertex set of a 2‐connected graph can be partitioned into two subsets of prescribed order, each inducing a connected subgraph. We prove that if we allow a small overlap between the sets, then the vertex set of a 2‐connected graph can be expressed as the union of three subsets of prescribed order, each inducing a connected subgraph.
ABSTRACT A connected graph is matching covered if it contains at least one edge and every edge lies in some perfect matching of . Lovász proved that every matching covered graph can be uniquely decomposed into a list of bricks (nonbipartite) and braces (bipartite) up to multiple edges; we let denote the number of bricks. An edge in a matching covered graph is removable if is also matching covered. Furthermore, a removable edge of a brick is ‐invariant if . Confirming a conjecture of Lovász, de Carvalho, Lucchesi, and Murty proved that every brick, distinct from , , and the Petersen graph, has a ‐invariant edge. A brick is near‐bipartite if it has a pair of edges such that is a bipartite matching covered graph. In this paper, strengthening the result of Zhang et al., we show that in a near‐bipartite brick with , every vertex of , except at most six vertices, is incident with at most two non‐‐invariant edges. Consequently, has at least ‐invariant edges, and all such graphs attaining this lower bound are presented.
Let be positive integers such that and be a tripartite graph with parts such that . Denote the edge densities of and by and , respectively. In this paper, we study edge density conditions for the existence of vertex-disjoint triangles in a tripartite graph. For we give an optimal condition in terms of densities for the existence of vertex-disjoint triangles in . We also give an optimal condition in terms of densities for the existence of a triangle-factor in .
In 1973, Chv & aacute;tal conjectured that there exists a constant such that every -tough graph on at least three vertices is Hamiltonian. This conjecture has inspired extensive research and has been verified for several special classes of graphs. Notably, Jung in 1978 proved that every 1-tough -free graph on at least three vertices is Hamiltonian. However, the problem remains challenging even when restricted to graphs with no induced boolean OR , the disjoint union of a path on four vertices and a one-vertex path. In 2013, Nikoghosyan conjectured that every 1-tough ( ) -free graph on at least three vertices is Hamiltonian. Later in 2015, Broersma remarked that "this question seems to be very hard to answer, even if we impose a higher toughness." Instead he posed the following question: "Is the general conjecture of Chv & aacute;tal's true for ( ) -free graphs?" We provide a positive answer to Broersma's question by establishing that every 23-tough ( ) -free graph on at least three vertices is Hamiltonian.
ABSTRACT In this paper, we are interested in 4‐colouring algorithms for graphs that do not contain an induced path on six vertices nor an induced bull, that is, the graph with vertex set and edge set . Such graphs are referred to as ‐free graphs. A graph is ‐ vertex‐critical if , and every proper induced subgraph of has . In the current paper, we investigate the structure of 5‐vertex‐critical ‐free graphs and show that there are only finitely many such graphs, thereby answering a question of Maffray and Pastor. A direct corollary of this is that there exists a polynomial‐time algorithm to decide if a ‐free graph is 4‐colourable such that this algorithm can also provide a certificate that can be verified in polynomial time and serves as a proof of 4‐colourability or non‐4‐colourability.
ABSTRACT A ‐ coloring of is a coloring of the edges of such that every ‐clique has at least distinct colors among its edges. The generalized Ramsey number is the minimum number of colors such that has a ‐coloring. Gomez‐Leos, Heath, Parker, Schweider and Zerbib recently proved . Here we prove an asymptotically matching upper bound.
ABSTRACT A graph is ‐connected if, for any mapping with , there exists a strongly connected orientation satisfying for any . It is known that ‐connected graphs are contractible configurations for the property of flow index strictly less than three. In this paper, we provide a complete characterization of graphic sequences that have an ‐connected realization: A graphic sequence has an ‐connected realization if and only if and . Consequently, every graphic sequence with has a realization with flow index strictly less than three. This supports the conjecture of Li, Thomassen, Wu and Zhang [European J. Combin., 70 (2018) 164‐177] that every 6‐edge‐connected graph has a flow index strictly less than three.
ABSTRACT A graph of order is said to be ‐factor‐critical () if the removal of any vertices results in a graph with a perfect matching. A ‐factor‐critical graph is minimal if is not ‐factor‐critical for any edge in . Favaron and Shi posed the conjecture that every minimal ‐factor‐critical graph is of minimum degree in 1998. In this paper, we confirm the conjecture for planar graphs.
The spanning tree packing number of a graph G , written by tau , is defined as the maximum number of edge-disjoint spanning trees. The study of tau is a classic problem in graph theory, and this parameter has important application in analyzing network robustness, routing optimization, and fault tolerance. Paul Seymour motivated the academia to establish the relationship between tau and eigenvalues of a connected graph. Let be the class of n -vertex graphs with minimum degree delta and let G denote the set of graphs of order n and size m admitting exactly k edge-disjoint spanning trees. Fan, Gu, and Lin [J. Graph Theory, 104 (2023) 697-711] gave a sufficient size condition, together with its spectral analogue, to ensure that a graph G is an element of satisfies tau >= k under the condition delta >= 2 k . It is interesting to see that these problems are still open under the condition k <= delta <= 2 k - 1 . In this paper, we fill this gap. Then we address an open problem posed by Fan, Gu, and Lin [J. Graph Theory, 104 (2023) 697-711]: Determine the graphs among G having maximum spectral radius. Furthermore, they conjectured that del is the unique graph among G attaining maximum spectral radius for n >= 4 and k >= 2 . We show that, for n >= with k >= 5 , del is the unique graph among G having maximum spectral radius. This resolves their open problem, and consequently disproves their conjecture.
Let be three integers such that and . Let be a -partite -uniform hypergraph with vertices in each class. Aharoni (2017) showed that if , then has a matching of size . In this paper, we give a stability result for 3-partite 3-uniform hypergraphs: if is a 3-partite 3-uniform hypergraph with vertices in each class, and contains no matching of size , then has a vertex cover of size . Our bound is also tight.
ABSTRACT Füredi and Gunderson showed that is achieved only on if . It is natural to study how far a ‐free graph is from being bipartite. If a graph and a graph have at most one vertex in common and there is no edge connecting and , then we call graph a suspension to graph with suspension point. Let be obtained by adding a suspension with 1 suspension point to . Let and . Ren, Wang, Wang, and Yang showed that if is an ‐vertex ‐free graph with , then and , and equalities hold if and only if for and . In this paper, we show that for integers with and , if is a ‐free ‐vertex graph with , then is obtained by adding suspensions to a ‘nearly balanced complete’ bipartite graph one by one and the number of vertices not in is no more than . Furthermore, the total number of vertices not in equals if and only if . Roughly speaking, we give a strong structural information for ‐free graph rather than the distance from being bipartite when if . In the proof, we introduce a new concept strong‐‐core which is the key that we can give a stronger structural stability result but a simpler proof.
ABSTRACT In an effort to understand the complexity of the maximum independent set problem, Chvátal introduced t‐perfect graphs. While a full characterization of this class remains open, important progress has been made for claw‐free graphs [Bruhn and Stein, Math. Program. 2012] and ‐free graphs [Bruhn and Fuchs, SIAM J. Discrete Math. 2017]. We take a further step by characterizing fork‐free t‐perfect graphs and showing that they are strongly t‐perfect and 3‐colorable. We also give polynomial‐time algorithms for recognizing and coloring fork‐free t‐perfect graphs.
Let denote the minimum number of colors needed to properly color the edges of a graph such that every 4-cycle is colored with four different colors. Very recently, Gy & aacute;rf & aacute;s et al. [3] proved that for a planar graph and for an outerplanar graph except and . They also conjectured that, when is large enough, every planar graph has and every outerplanar graph has . Let be a planar graph. In this paper, we show the following results: (1) ; (2) if ; (3) if ; (4) if is outerplanar and . Results (3) and (4) confirm the conjectures of Gy & aacute;rf & aacute;s et al. [3].
