Graphs H1 Research Articles

Ramsey-Turán type results for matchings in edge colored graphs

Let G1,G2,…,Gc be arbitrary graphs. By G→(G1,G2,…,Gc), we mean if the edges of G are partitioned into c disjoint color classes giving c graphs H1,H2,…,Hc, then at least one Hi has a subgraph isomorphic to Gi. The Ramsey number R(G1,G2,…,Gc) is the smallest positive integer n such that Kn→(G1,G2,…,Gc). In this paper, we extend the result of Cockayne and Lorimer on the Ramsey number of stripes allowing host graphs with the corresponding Ore-type condition: If t≥1 and n1,n2,…,nc, n1=max⁡{n1,n2,…,nc}, are positive integers and G is a graph with at least max⁡{t,n1}+∑i=1c(ni−1)+1 vertices such that for each pair of non-adjacent vertices, the sum of the number of their non-neighbors is at most 2max⁡{t,n1}−1, then G→(n1K2,n2K2,…,ncK2). Consequently, we prove that if G is a graph with chromatic number at least max⁡{t,n1}+∑i=1c(ni−1)+1, then G→(St,n1K2,n2K2,…,ncK2), where St is the star graph with t edges. In addition, for given graph G, the Ramsey closure of G, denoted by C(G), is defined and we prove that if G is a graph with at least max⁡{t,n1}+∑i=1c(ni−1)+1 vertices, then G→(n1K2,n2K2,…,ncK2) if and only if C(G)→(n1K2,n2K2,…,ncK2). Thus, a condition that forces C(G)→(n1K2,n2K2,…,ncK2) also forces G→(n1K2,n2K2,…,ncK2). As a result, a condition is provided on the degree sequence of a graph G such that under this condition G→(n1K2,n2K2,…,ncK2).Further, this article provides a sharp bound for the maximum number of edges in a simple graph G such that G↛(n1K2,n2K2,…,ncK2). We also establish the uniqueness of such extremal graphs. This result provides a far-reaching generalization of an important classical result of Erdős and Gallai and also giving a new proof for the graph case of the conjecture of Erdős dating back to 1965, known as the Erdős' matching conjecture.

Vertex-critical [formula omitted]-free graphs

Given two graphs H1 and H2, a graph G is (H1,H2)-free if it contains no induced subgraph isomorphic to H1 or H2. A Pt is the path on t vertices. A chair is a P4 with an additional vertex adjacent to one of the middle vertices of the P4. A graph G is k-vertex-critical if G has chromatic number k but every proper induced subgraph of G has chromatic number less than k. In this paper, we prove that there are only finitely many 5-vertex-critical (P5,chair)-free graphs.

Multicolor bipartite Ramsey numbers for quadrilaterals and stars

For bipartite graphs H1,…,Hμ, μ≥2, the μ-color bipartite Ramsey number, denoted by Rb(H1,…,Hμ), is the least positive integer N such that if we arbitrarily color the edges of a complete bipartite graph KN,N with μ colors, then it contains a monochromatic copy of Hi in color i for some i, 1≤i≤μ. Let C4 and K1,n be a quadrilateral and a star on n+1 vertices, respectively. In this paper, we show that the (μ+1)-color bipartite Ramsey number Rb(C4,…,C4,K1,n)≤n+⌈12μ2(4n+μ2+2μ−7)+4⌉+μ2+μ2−1. Moreover, using algebraic methods, we construct Ramsey graphs or near Ramsey graphs and determine infinitely many values of Rb(C4,…,C4,K1,n), which reach the upper bound if μ=1,2 and are at most ⌊μ2⌋ less than the upper bound if μ≥3.

A path forward: Tropicalization in extremal combinatorics

Many important problems in extremal combinatorics can be stated as proving a pure binomial inequality in graph homomorphism numbers, i.e., proving that hom(H1,G)a1⋯hom(Hk,G)ak≥hom(Hk+1,G)ak+1⋯hom(Hm,G)am holds for some fixed graphs H1,…,Hm and all graphs G. One prominent example is Sidorenko's conjecture. For a fixed collection of graphs U={H1,…,Hm}, the exponent vectors of valid pure binomial inequalities in graphs of U form a convex cone. We compute this cone for several families of graphs including complete graphs, even cycles, stars and paths; the latter is the most interesting and intricate case that we compute. In all of these cases, we observe a tantalizing polyhedrality phenomenon: the cone of valid pure binomial inequalities is actually rational polyhedral, and therefore all valid pure binomial inequalities can be generated from the finite collection of exponent vectors of the extreme rays. Using the work of Kopparty and Rossman ([17]), we show that the cone of valid inequalities is indeed rational polyhedral when all graphs Hi are series-parallel and chordal, and we conjecture that polyhedrality holds for any finite collection U. We demonstrate that the polyhedrality phenomenon also occurs in matroids and simplicial complexes. Our description of the inequalities for paths involves a generalization of the Erdős-Simonovits conjecture recently proved in its original form in [29] and a new family of inequalities not observed previously. We also solve an open problem of Kopparty and Rossman on the homomorphism domination exponent of paths. One of our main tools is tropicalization, a well-known technique in complex algebraic geometry, first applied in extremal combinatorics in [3]. We prove several results about tropicalizations which may be of independent interest.

Bounds for two multicolor Ramsey numbers concerning quadrilaterals

For k given graphs H1,…,Hk, k≥2, the k-color Ramsey number, denoted by R(H1,…,Hk), is the smallest integer N such that if we arbitrarily color the edges of a complete graph of order N with k colors, then it always contains a monochromatic copy of Hi colored with i, for some 1≤i≤k. Let Cm be a cycle of length m, K1,n a star of order n+1 and Wn a wheel of order n+1. In this paper, by using algebraic and probabilistic methods, we first give two lower bounds for (k+1)-color Ramsey number R(C4,…,C4,K1,n) for some special n, which shows the upper bound due to Zhang et al. (2019) is tight in some sense, and then establish a general lower bound for R(C4,…,C4,K1,n) in terms of n and k, which extends the classical result of Burr et al. (1989). Moreover, we show that R(C4,…,C4,K1,n)=R(C4,…,C4,Wn) for sufficiently large n.

Common multiples of paths and stars with complete graphs

A graph G is a common multiple of two graphs H1 and H2 if there exists a decomposition of G into edge-disjoint copies of H1 and also a decomposition of G into edge-disjoint copies of H2. If G is a common multiple of H1 and H2 and G has q edges, then we call G a (q, H1, H2) graph. Our paper deals with the following question: ''Given two graphs H1 and H2, for which values of q does there exist a (q, H1, H2) graph?'' when H1 is either a path or a star with 3 or 4 edges and H2 is a complete graph.

New bounds for Ramsey numbers [formula omitted

Let R(H1,H2) denote the Ramsey number for the graphs H1,H2, and let Jk be Kk−e. We present algorithms which enumerate all circulant and block-circulant Ramsey graphs for different types of graphs, thereby obtaining several new lower bounds on Ramsey numbers including: 49≤R(K3,J12), 36≤R(J4,K8), 43≤R(J4,J10), 52≤R(K4,J8), 37≤R(J5,J6), 43≤R(J5,K6), 65≤R(J5,J7). We also use a gluing strategy to derive a new upper bound on R(J5,J6). With both strategies combined, we prove the value of two Ramsey numbers: R(J5,J6)=37 and R(J5,J7)=65. We also show that the 64-vertex extremal Ramsey graph for R(J5,J7) is unique. Furthermore, our algorithms also allow the establishment of new lower bounds and exact values on Ramsey numbers involving wheel graphs and complete bipartite graphs, including: R(W7,W4)=21, R(W7,W7)=19, R(K3,4,K3,4)=25, and R(K3,5,K3,5)=33.

Gallai–Ramsey numbers for multiple triangles

A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edges colored differently. A Gallai k-coloring is a Gallai coloring that uses at most k colors. Given an integer k≥2, and graphs H1,H2,…,Hk, the Gallai-Ramsey number GR(H1,H2,…,Hk) is the least positive integer N such that every Gallai k-coloring of the complete graph KN contains a monochromatic copy of Hi in color i for some i∈{1,2,…,k}. Let mG denote the union of m disjoint copies of G. This paper studies the Gallai–Ramsey number GR(t1K3,…,tkK3) for all integers t1≥t2≥⋯≥tk≥1. The case t1=t2=⋯=tk=1 was obtained by Chung and Graham (1983). In this paper, we obtain a general bound of GR(t1K3,…,tkK3) for all t1≥t2≥⋯≥tk≥1, and prove that both the upper and lower bounds can be attained for some ti, i∈{1,…,k}.

K-Critical graphs in P5-free graphs

Given two graphs H1 and H2, a graph G is (H1,H2)-free if it contains no induced subgraph isomorphic to H1 or H2. Let Pt be the path on t vertices. A graph G is k-vertex-critical if G has chromatic number k but every proper induced subgraph of G has chromatic number less than k. The study of k-vertex-critical graphs for graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there is a polynomial-time algorithm to decide if a graph in the class is (k−1)-colorable.In this paper, we initiate a systematic study of the finiteness of k-vertex-critical graphs in subclasses of P5-free graphs. Our main result is a complete classification of the finiteness of k-vertex-critical graphs in the class of (P5,H)-free graphs for all graphs H on 4 vertices. To obtain the complete dichotomy, we prove the finiteness for four new graphs H using various techniques – such as Ramsey-type arguments and the dual of Dilworth's Theorem – that may be of independent interest.

On characterizing the critical graphs for matching Ramsey numbers

Given simple graphs H1,H2,…,Hc, the Ramsey number r(H1,H2,…,Hc) is the smallest positive integer n such that every edge-colored Kn with c colors contains a subgraph in color i isomorphic to Hi for some i∈{1,2,…,c}. The critical graphs for r(H1,H2,…,Hc) are edge-colored complete graphs on r(H1,H2,…,Hc)−1 vertices with c colors which contain no subgraphs in color i isomorphic to Hi for any i∈{1,2,…,c}. For n1≥n2≥⋯≥nc≥1, Cockayne and Lorimer (1975) showed that r(n1K2,n2K2,…,ncK2)=n1+1+∑i=1c(ni−1), in which niK2 is a matching of size ni. Using the Gallai–Edmonds Theorem, we characterized all the critical graphs for r(n1K2,n2K2,…,ncK2), implying a new proof for this Ramsey number.

Ramsey properties of randomly perturbed graphs: cliques and cycles

AbstractGiven graphs H1, H2, a graph G is (H1, H2) -Ramsey if, for every colouring of the edges of G with red and blue, there is a red copy of H1 or a blue copy of H2. In this paper we investigate Ramsey questions in the setting of randomly perturbed graphs. This is a random graph model introduced by Bohman, Frieze and Martin [8] in which one starts with a dense graph and then adds a given number of random edges to it. The study of Ramsey properties of randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali [30] in 2006; they determined how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (K3, Kt) -Ramsey (for t ≽ 3). They also raised the question of generalizing this result to pairs of graphs other than (K3, Kt). We make significant progress on this question, giving a precise solution in the case when H1 = Ks and H2 = Kt where s, t ≽ 5. Although we again show that one requires polynomially fewer edges than in the purely random graph, our result shows that the problem in this case is quite different to the (K3, Kt) -Ramsey question. Moreover, we give bounds for the corresponding (K4, Kt) -Ramsey question; together with a construction of Powierski [37] this resolves the (K4, K4) -Ramsey problem.We also give a precise solution to the analogous question in the case when both H1 = Cs and H2 = Ct are cycles. Additionally we consider the corresponding multicolour problem. Our final result gives another generalization of the Krivelevich, Sudakov and Tetali [30] result. Specifically, we determine how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (Cs, Kt) -Ramsey (for odd s ≽ 5 and t ≽ 4).To prove our results we combine a mixture of approaches, employing the container method, the regularity method as well as dependent random choice, and apply robust extensions of recent asymmetric random Ramsey results.

On minimal Ramsey graphs and Ramsey equivalence in multiple colours

AbstractFor an integerq⩾ 2, a graphGis calledq-Ramsey for a graphHif everyq-colouring of the edges ofGcontains a monochromatic copy ofH. IfGisq-Ramsey forHyet no proper subgraph ofGhas this property, thenGis calledq-Ramsey-minimal forH. Generalizing a statement by Burr, Nešetřil and Rödl from 1977, we prove that, forq⩾ 3, ifGis a graph that is notq-Ramsey for some graphH, thenGis contained as an induced subgraph in an infinite number ofq-Ramsey-minimal graphs forHas long asHis 3-connected or isomorphic to the triangle. For suchH, the following are some consequences.For 2 ⩽r<q, everyr-Ramsey-minimal graph forHis contained as an induced subgraph in an infinite number ofq-Ramsey-minimal graphs forH.For everyq⩾ 3, there areq-Ramsey-minimal graphs forHof arbitrarily large maximum degree, genus and chromatic number.The collection$\{\mathcal M_q(H) \colon H \text{ is 3-connected or } K_3\}$forms an antichain with respect to the subset relation, where$\mathcal M_q(H)$denotes the set of all graphs that areq-Ramsey-minimal forH.We also address the question of which pairs of graphs satisfy$\mathcal M_q(H_1)=\mathcal M_q(H_2)$, in which caseH1andH2are calledq-equivalent. We show that two graphsH1andH2areq-equivalent for evenqif they are 2-equivalent, and that in generalq-equivalence for someq⩾ 3 does not necessarily imply 2-equivalence. Finally we indicate that for connected graphs this implication may hold: results by Nešetřil and Rödl and by Fox, Grinshpun, Liebenau, Person and Szabó imply that the complete graph is not 2-equivalent to any other connected graph. We prove that this is the case for an arbitrary number of colours.

On the power domination number of corona product and join graphs

A set D of vertices of graph G is called a dominating set if every vertex is adjacent to some vertex . A set D to be a power dominating set of a graph if every vertex and every edge in the system are monitored by the set D. The power domination number γP (G) of a graph G is the minimum cardinality of a power dominating set of G. In this paper, we analyze the power domination number of corona graphs and join graphs. The corona product of two graphs G1 and G2 denoted by is defined as the graph G obtained by taking one copy of G1 and copies of G2, and then joined by an edge the i’th vertex of G1 to every vertex in the i’th copy of G2. The join of two graphs H1 and H2 is a graph formed from disjoint copies of H1 and H2 by connecting every vertex of H1 to every vertex of H2. Join graph H1 and H2 denoted by H1 + H2. The results show that the power domination number of some corona product and join graphs attain the lower bound.

On r-dynamic chromatic number of coronation of order two of any graphs with path graph

In this study we assume every graphs be a simple, finite, nontrivial, connected and undirected graphs with vertex set V(H), edge set E(H) and no isolated vertex. An r-dynamic coloring of a graph H is a proper k-coloring of graph H such that the neighbors of any vertex u accept at least min{r, d(u)} different colors. The r-dynamic chromatic number is the minimum k such that graph H has an r-dynamic k-coloring, it is shown by χr(H). In this research, we interpret the r-dynamic chromatic number of graph corona of order two, denoted by H1⊙2 H2. The corona of two graphs H1 and H2 is a graph H1 ⊙ H2 formed from one copy of H1 and |V(H1)| copies of H2 where the ith vertex of H1 is adjacent to every vertex in the ith copy of H2. For any integer l ≥ 2, we establish the graph H1 ⊙l H2 recursively from H1 ⊙ H2 as H1 ⊙l H2 = (H1 ⊙l−1 H2) ⊙ H2. Graph H1 ⊙l H2 is also named as l − corona product of H1 and H2. Xr = (Pn ⊙2 H2) and H2 is path graph Pm, star graph Sm, complete graph Km and fan graph Fm. In this study we will determine the lower bound of r-dynamic chromatic number of corona order two of graphs.

(H1, H2)-supermagic labelings for some shackles of connected graphs H1 and H2

Let H1 and H2 be two graphs. A simple graph G = (V(G),E(G)) admits an (H1, H2)-covering, if every edge in E(G) belongs to at least one subgraph of G isomorphic to H1 or H2. The graph G is called (H1, H2)-magic, if there are two fixed positive integers ki and k2, and a bijective function f : V(G) ∪ E(G) → {1, 2,…, |V(G)| + |E(G)|} suchthat ∑v∈V(H′) f(v) + ∑e∈E(H′) f(e) = k1 and ∑v∈V(H″) f(v) + ∑e∈E(H″) f(e) = k2, for every subgraph H′ = (V(H′), E(H′)) of G isomorphic to H1 and for every subgraph H″ = (V(H), E(H)) of G isomorphic to H2. Moreover, it is said to be super (H1,H2)-magic, if f(V(G)) = {1, 2,…, |V(g)|}.This paper aims to study an (H1, H2)-supermagic labelings for some shackles of connected graphs H1 and H2 such as cycle, flower, and prism graph. A shackle of G1, G2, G3,…, Gk denoted by shack(G1, G2, G3,…, Gk) is a graph constructed from nontrivial connected and ordered graphs suchthat for every i and j ∈ [1, k] with |i − j| > 2, Gi and Gj have no common vertex, and for every i ∈ [1, k − 1], Gi and Gi+1 share exactly one common vertex, called linkage vertex, where the k − 1 linkage vertices are all distinct. In case Gi is isomorphic to H1 for odd i and Gi is isomorphic to H2 for even i, we denote such shackle by shack(H1, H2, k). We give a sufficient condition for some shack(H1, H2, k) being (H1, H2)-supermagic for even k.

Critical [formula omitted]-free graphs

Given two graphs H1 and H2, a graph is (H1,H2)-free if it contains no induced subgraph isomorphic to H1 or H2. Let Pt and Ct be the path and the cycle on t vertices, respectively. A banner is the graph obtained from a C4 by adding a new vertex and making it adjacent to exactly one vertex of the C4. In this paper, we show that there are finitely many k-critical (P6,banner)-free graphs for k=4 and k=5. For k=4, we characterize all such graphs. Our results generalize previous results on k-critical (P6,C4)-free graphs for k=4 and k=5.

On three-color Ramsey numbers R(C4,K1,m,Pn)

For given graphs H1,...,Hk, k≥2, the k-color Ramsey number R(H1,...,Hk) is the smallest integer N such that every k-coloring of the edges of a complete graph KN contains a monochromatic copy of Hi colored in i, for some i with 1≤i≤k. Let Cℓ,K1,m and Pn denote a cycle of length ℓ, a star of order m+1 and a path of order n, respectively. In this paper, it is shown that R(C4,K1,m,Pn)≤m+n−1+⌈m+n−2⌉ for all m,n≥2 and R(C4,K1,m,Pn)≤m+n−2+⌈m+n−2⌉ if m+n=ℓ2+3 and ℓ≥1. Moreover, by discussing the local structure of the polarity graph whose vertices are points in the projective plane over Galois fields, we show that the two upper bounds can be attained for some special m and n. These results also extend some known results on R(C4,K1,m) obtained by Parsons in 1975 and by Zhang et al. recently.

Saturation numbers for Ramsey-minimal graphs

Given graphs H1,…,Ht, a graph G is (H1,…,Ht)-Ramsey-minimal if every t-coloring of the edges of G contains a monochromatic Hi in color i for some i∈{1,…,t}, but any proper subgraph of G does not possess this property. We define Rmin(H1,…,Ht) to be the family of (H1,…,Ht)-Ramsey-minimal graphs. A graph G is Rmin(H1,…,Ht)-saturated if no element of Rmin(H1,…,Ht) is a subgraph of G, but for any edge e in G¯, some element of Rmin(H1,…,Ht) is a subgraph of G+e. We define sat(n,Rmin(H1,…,Ht)) to be the minimum number of edges over all Rmin(H1,…,Ht)-saturated graphs on n vertices. In 1987, Hanson and Toft conjectured that sat(n,Rmin(Kk1,…,Kkt))=(r−2)(n−r+2)+r−22 for n≥r, where r=r(Kk1,…,Kkt) is the classical Ramsey number for complete graphs. The first non-trivial case of Hanson and Toft’s conjecture for sufficiently large n was settled in 2011, and is so far the only settled case. Motivated by Hanson and Toft’s conjecture, we study the minimum number of edges over all Rmin(K3,Tk)-saturated graphs on n vertices, where Tk is the family of all trees on k vertices. We show that for n≥18, sat(n,Rmin(K3,T4))=⌊5n∕2⌋. For k≥5 and n≥2k+(⌈k∕2⌉+1)⌈k∕2⌉−2, we obtain an asymptotic bound for sat(n,Rmin(K3,Tk)) by showing that 32+12k2n−c≤sat(n,Rmin(K3,Tk))≤32+12k2n+C, where c=12k2+32k−2 and C=2k2−6k+32−k2k−12k2−1.

Super (a*, d*)-ℋ-antimagic total covering of second order of shackle graphs

Let H be a simple and connected graph. A shackle of graph H, denoted by G = shack(H, v, n), is a graph G constructed by non-trivial graphs H1, H2, …, Hn such that, for every 1 ≤ s, t ≤ n, Hs and Ht have no a common vertex with |s − t| ≥ 2 and for every 1 ≤ i ≤ n − 1, Hi and Hi+1 share exactly one common vertex v, called connecting vertex, and those k − 1 connecting vertices are all distinct. The graph G is said to be an (a*, d*)-H-antimagic total graph of second order if there exist a bijective function f : V(G) ∪ E(G) → {1, 2, …, |V(G)| + |E(G)|} such that for all subgraphs isomorphic to H, the total H-weights form an arithmetic sequence of second order of , where a* and d* are positive integers and n is the number of all subgraphs isomorphic to H. An (a*, d*)-H-antimagic total labeling of second order f is called super if the smallest labels appear in the vertices. In this paper, we study a super (a*, d*)-H antimagic total labeling of second order of G = shack(H, v, n) by using a partition technique of second order.

A generalized shackle of any graph H admits a super H-antimagic total labeling

Let H be a simple and connected graph. A shackle of graph H, denoted by G = shack(H, v, n), is a graph G constructed by non-trivial graphs H1, H2, …, Hn such that, for every 1 ≤ s, t ≤ n, Hs and Ht have no a common vertex with |s − t| ≥ 2 and for every 1 ≤ i ≤ n − 1, Hi and Hi+1 share exactly one common vertex v, called connecting vertex, and those k − 1 connecting vertices are all distinct. By a generalized shackle of graph, we mean the graph G = shack(H, v, n) by replacing the connecting vertex by any subgraph K ⊂ H and we denote such a graph as G = shack(H, K ⊂ H, n). Graph G = gshack(H, K ⊂ H, n) admits a H-covering, if every edge in E(G) belongs to at least one subgraph of G isomorphic to a given graph H. A graph G is an (a, d)−H-antimagic total graph if there exists a bijective function f : V(G) ⋃ E(G) → {1, 2, …,|V(G)| + |E(G)|} such that for all subgraphs isomorphic to H, the total H-weights w(H) = Σv∈V(H) f(v) + Σe∈E(H) f(e) form an arithmetic sequence {a, a + d, a + 2d, …, a + (n − 1)d}, where a and d are positive integers and n is the number of all subgraphs isomorphic to H. If such a function exists then f is called an (a, d)−H-antimagic total labeling of G. An (a, d)−H-antimagic total labeling f is called super if f : V (G) → {1, 2, …, |V(G)|}. In this paper, we study a super (a, d)−H antimagic total labeling of G = gshack(H, e ∈ H, n) by using a partition technique. The result shows that there exist a super(a, d)−H antimagic total labeling for almost feasible difference d up to the determined upper bound.