Game-perfect graphs

A graph coloring game introduced by Bodlaender (Int J Found Comput Sci 2:133–147, 1991) as coloring construction game is the following. Two players, Alice and Bob, alternately color vertices of a given graph G with a color from a given color set C, so that adjacent vertices receive distinct colors. Alice has the first move. The game ends if no move is possible any more. Alice wins if every vertex of G is colored at the end, otherwise Bob wins. We consider two variants of Bodlaender’s graph coloring game: one (A) in which Alice has the right to have the first move and to miss a turn, the other (B) in which Bob has these rights. These games define the A-game chromatic number resp. the B-game chromatic number of a graph. For such a variant g, a graph G is g-perfect if, for every induced subgraph H of G, the clique number of H equals the g-game chromatic number of H. We determine those graphs for which the game chromatic numbers are 2 and prove that the triangle-free B-perfect graphs are exactly the forests of stars, and the triangle-free A-perfect graphs are exactly the graphs each component of which is a complete bipartite graph or a complete bipartite graph minus one edge or a singleton. From these results we may easily derive the set of triangle-free game-perfect graphs with respect to Bodlaender’s original game. We also determine the B-perfect graphs with clique number 3. As a general result we prove that complements of bipartite graphs are A-perfect.      

Some observations on maximum weight stable sets in certain P5-free graphs

The maximum weight stable set problem (MWS) is the weighted version of the maximum stable set problem (MS), which is NP-hard. The class of P₅-free graphs – i.e., graphs with no induced path of five vertices – is the unique minimal class, defined by forbidding a single connected subgraph, for which the computational complexity of MS is an open question. At the same time, it is known that MS can be efficiently solved for (_P5,F)$(P_5\text{,}F)$-free graphs, where F is any graph of five vertices different to a C₅. In this paper we introduce some observations on P₅-free graphs, and apply them to introduce certain subclasses of such graphs for which one can efficiently solve MWS. That extends or improves some known results, and implies – together with other known results – that MWS can be efficiently solved for (_P5,F)$(P_5\text{,}F)$-free graphs where F is any graph of five vertices different to a C₅.     

The binding number of a graph and its cliques

We consider the binding numbers of K_r-free graphs, and improve the upper bounds on the binding number which force a graph to contain a clique of order r. For the case r=4, we provide a construction for K₄-free graphs which have a larger binding number than the previously known constructions. This leads to a counterexample to a conjecture by Caro regarding the neighborhoods of independent sets.     

NP-hardness of the recognition of coordinated graphs

A graph G is coordinated if the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color is equal to the maximum number of cliques of H with a common vertex, for every induced subgraph H of G. In previous works, polynomial time algorithms were found for recognizing coordinated graphs within some classes of graphs. In this paper we prove that the recognition problem for coordinated graphs is NP-hard, and it is NP-complete even when restricted to the class of {gem, C ₄, odd hole}-free graphs with maximum degree four, maximum clique size three and at most three cliques sharing a common vertex. F.J. Soulignac is partially supported by UBACyT Grant X184, Argentina and CNPq under PROSUL project Proc. 490333/2004-4, Brazil.     

On lower bounds for the chromatic number in terms of vertex degree

In this paper we discuss the existence of lower bounds for the chromatic number of graphs in terms of the average degree or the coloring number of graphs. We obtain a lower bound for the chromatic number of K_1,t-free graphs in terms of the maximum degree and show that the bound is tight. For any tree T, we obtain a lower bound for the chromatic number of any K_2,t-free and T-free graph in terms of its average degree. This answers affirmatively a modified version of Problem 4.3 in [T.R. Jensen, B. Toft, Graph Coloring Problems, Wiley, New York, 1995]. More generally, we discuss δ-bounded families of graphs and then we obtain a necessary and sufficient condition for a family of graphs to be a δ-bounded family in terms of its induced bipartite Turán number. Our last bound is in terms of forbidden induced even cycles in graphs; it extends a result in [S.E. Markossian, G.S. Gasparian, B.A. Reed, β-perfect graphs, J. Combin. Theory Ser. B 67 (1996) 1–11].     

Bounds for the b-chromatic number of some families of graphs

In this paper we obtain some upper bounds for the b-chromatic number of K_1,s-free graphs, graphs with given minimum clique partition and bipartite graphs. These bounds are given in terms of either the clique number or the chromatic number of a graph or the biclique number for a bipartite graph. We show that all the bounds are tight.     

Paired-Domination in <Emphasis Type="Italic">P</Emphasis><Subscript>5</Subscript>-Free Graphs

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by , is the minimum cardinality of a paired-dominating set of G. In [1], the authors gave tight bounds for paired-dominating sets of generalized claw-free graphs. Yet, the critical cases are not claws but subdivided stars. We here give a bound for graphs containing no induced P ₅, which seems to be the critical case.     

A generalization of perfect graphs—i-perfect graphs

Let i be a positive integer. We generalize the chromatic number _X(G) of G and the clique number o(G) of G as follows: The i-chromatic number of G, denoted by _X(G), is the least number k for which G has a vertex partition V₁, V₂,…, V_k such that the clique number of the subgraph induced by each V_j, 1 ≤ j ≤ k, is at most i. The i-clique number, denoted by o_i(G), is the i-chromatic number of a largest clique in G, which equals [o(G/i]. Clearly _X1(G) = _X(G) and o₁(G) = o(G). An induced subgraph G′ of G is an i-transversal iff o(G′) = i and o(G − G′) = o(G) − i. We generalize the notion of perfect graphs as follows: (1) A graph G is i-perfect iff _Xi(H) = o_i(H) for every induced subgraph H of G. (2) A graph G is perfectly i-transversable iff either o(G) ≤ i or every induced subgraph H of G with o(H) > i contains an i-transversal of H. We study the relationships among i-perfect graphs and perfectly i-transversable graphs. In particular, we show that 1-perfect graphs and perfectly 1-transversable graphs both coincide with perfect graphs, and that perfectly i-transversable graphs form a strict subset of i-perfect graphs for every i ≥ 2. We also show that all planar graphs are i-perfect for every i ≥ 2 and perfectly i-transversable for every i ≥ 3; the latter implies a new proof that planar graphs satisfy the strong perfect graph conjecture. We prove that line graphs of all triangle-free graphs are 2-perfect. Furthermore, we prove for each i greater than or equal to2, that the recognition of i-perfect graphs and the recognition of perfectly i-transversable graphs are intractable and not likely to be in co-NP. We also discuss several issues related to the strong perfect graph conjecture. © 1996 John Wiley & Sons, Inc.     

On αrγs(k)-perfect graphs

For some integer k0 and two graph parameters π and τ, a graph G is called πτ(k)-perfect, if π(H)−τ(H)k for every induced subgraph H of G. For r1 let α_r and γ_r denote the r-(distance)-independence and r-(distance)-domination number, respectively. In (J. Graph Theory 32 (1999) 303–310), I. Zverovich gave an ingenious complete characterization of α₁γ₁(k)-perfect graphs in terms of forbidden induced subgraphs. In this paper we study α_rγ_s(k)-perfect graphs for r,s1. We prove several properties of minimal α_rγ_s(k)-imperfect graphs. Generalizing Zverovich's main result in (J. Graph Theory 32 (1999) 303–310), we completely characterize α_2r−1γ_r(k)-perfect graphs for r1. Furthermore, we characterize claw-free α₂γ₂(k)-perfect graphs.     

Upper domination and upper irredundance perfect graphs

Let β(G), Γ(G) and IR(G) be the independence number, the upper domination number and the upper irredundance number, respectively. A graph G is called Γ-perfect if β(H) = Γ(H), for every induced subgraph H of G. A graph G is called IR-perfect if Γ(H) =IR(H), for every induced subgraph H of G. In this paper, we present a characterization of Γ-perfect graphs in terms of a family of forbidden induced subgraphs, and show that the class of Γ-perfect graphs is a subclass of IR-perfect graphs and that the class of absorbantly perfect graphs is a subclass of Γ-perfect graphs. These results imply a number of known theorems on Γ-perfect graphs and IR-perfect graphs. Moreover, we prove a sufficient condition for a graph to be Γ-perfect and IR-perfect which improves a known analogous result.     

Conditions for -graphic sequences to be potentially -graphic

An r-graph is a loopless undirected graph in which no two vertices are joined by more than r edges. An r-complete graph on m+1 vertices, denoted by , is an r-graph on m+1 vertices in which each pair of vertices is joined by exactly r edges. A non-increasing sequence π=(d₁,d₂,…,d_n) of nonnegative integers is said to be r-graphic if it is realizable by an r-graph on n vertices. An r-graphic sequence π is said to be potentially -graphic if it has a realization containing as a subgraph. In this paper, some conditions for r-graphic sequences to be potentially -graphic are given. These are generalizations from 1-graphs to r-graphs of four theorems due to Rao [A.R. Rao, The clique number of a graph with given degree sequence, in: A.R. Rao (Ed.), Proc. Symposium on Graph Theory, in: I.S.I. Lecture Notes Series, vol. 4, MacMillan and Co. India Ltd., (1979), 251–267; A.R. Rao, An Erdös-Gallai type result on the clique number of a realization of a degree sequence (unpublished)] and Kézdy and Lehel [A.E. Kézdy, J. Lehel, Degree sequences of graphs with prescribed clique size, in: Y. Alavi et al., (Eds.), in: Combinatorics, Graph Theory, and Algorithms, vol. 2, New Issues Press, Kalamazoo Michigan, 1999, 535–544].     

On the Maximum Number of Cliques in a Graph

A clique is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for the following graph classes: (1) graphs with n vertices and m edges; (2) graphs with n vertices, m edges, and maximum degree Δ; (3) d-degenerate graphs with n vertices and m edges; (4) planar graphs with n vertices and m edges; and (5) graphs with n vertices and no K₅-minor or no K_3,3-minor. For example, the maximum number of cliques in a planar graph with n vertices is 8(n − 2). Research supported by a Marie Curie Fellowship of the European Community under contract 023865, and by the projects MCYT-FEDER BFM2003-00368 and Gen. Cat 2001SGR00224.     

Linearity of Zp[[t]]-Perfect Groups

Let G be a _p[[t]]-standard group of level 1. Then G is _p[[t]]-perfect if its lower central series is given by powers of the maximal ideal (p, t), i.e. if _n(G) = G((p,t)ⁿ). We prove that a _p[[t]]-perfect group is linear by imitating the proof that a _p[[t]]-standard group is linear.     

Graphs satisfying inequality θ(G)θ(G)

In this paper, we study the edge clique cover number of squares of graphs. More specifically, we study the inequality θ(G²)θ(G) where θ(G) is the edge clique cover number of a graph G. We show that any graph G with at most θ(G) vertices satisfies the inequality. Among the graphs with more than θ(G) vertices, we find some graphs violating the inequality and show that dually chordal graphs and power-chordal graphs satisfy the inequality. Especially, we give an exact formula computing θ(T²) for a tree T.     

On hamiltonicity of <Emphasis Type="Italic">P</Emphasis><Subscript>3</Subscript>-dominated graphs

We introduce a new class of graphs which we call P ₃-dominated graphs. This class properly contains all quasi-claw-free graphs, and hence all claw-free graphs. Let G be a 2-connected P ₃-dominated graph. We prove that G is hamiltonian if α(G ²) ≤ κ(G), with two exceptions: K _2,3 and K _1,1,3. We also prove that G is hamiltonian, if G is 3-connected and |V(G)| ≤ 5δ(G) − 5. These results extend known results on (quasi-)claw-free graphs. This paper was completed when both authors visited the Center for Combinatorics, Nankai University, Tianjin. They gratefully acknowledge the hospitality and support of the Center for Combinatorics and Nankai University. The work of E.Vumar is sponsored by SRF for ROCS, REM.     

图的符号团边控制数

本文研究了图的符号团边控制数的问题.利用鸽巢原理,获得了图K_n∨P_m和K_n∨C_m的符号团边控制数,推广了已有的结果.     

Conditions on Ramsey Nonequivalence

Given a graph H , a graph G is called a Ramsey graph of H if there is a monochromatic copy of H in every coloring of the edges of G with two colors. Two graphs G , H are called Ramsey equivalent if they have the same set of Ramsey graphs. Fox et al. (J Combin Theory Ser B 109 (2014), 120–133) asked whether there are two nonisomorphic connected graphs that are Ramsey equivalent. They proved that a clique is not Ramsey equivalent to any other connected graph. Results of Ne?et?il et al. showed that any two graphs with different clique number (Combinatorica 1(2) (1981), 199–202) or different odd girth (Comment Math Univ Carolin 20(3) (1979), 565–582) are not Ramsey equivalent. These are the only structural graph parameters we know that “distinguish” two graphs in the above sense. This article provides further supportive evidence for a negative answer to the question of Fox et al. by claiming that for wide classes of graphs, the chromatic number is a distinguishing parameter. In addition, it is shown here that all stars and paths and all connected graphs on at most five vertices are not Ramsey equivalent to any other connected graph. Moreover, two connected graphs are not Ramsey equivalent if they belong to a special class of trees or to classes of graphs with clique‐reduction properties.     

Characterizations for co-graphs defined by restricted NLC-width or clique-width operations

In this paper we characterize subclasses of co-graphs defined by restricted NLC-width operations and subclasses of co-graphs defined by restricted clique-width operations.We show that a graph has NLCT-width 1 if and only if it is (C₄,P₄)-free. Since (C₄,P₄)-free graphs are exactly trivially perfect graphs, the set of graphs of NLCT-width 1 is equal to the set of trivially perfect graphs, and a recursive definition for trivially perfect graphs follows. Further we show that a graph has linear NLC-width 1 if and only if is (C₄,P₄,2K₂)-free. This implies that the set of graphs of linear NLC-width 1 is equal to the set of threshold graphs.We also give forbidden induced subgraph characterizations for co-graphs defined by restricted clique-width operations using P₄, 2K₂, and co-2P₃.     

Several parameters of generalized Mycielskians

The generalized Mycielskians (also known as cones over graphs) are the natural generalization of the Mycielski graphs (which were first introduced by Mycielski in 1955). Given a graph G and any integer m?0, one can transform G into a new graph μ_m(G), the generalized Mycielskian of G. This paper investigates circular clique number, total domination number, open packing number, fractional open packing number, vertex cover number, determinant, spectrum, and biclique partition number of μ_m(G).     

A note on Ki-perfect graphs

Ki-perfect graphs are a special instance of F - G perfect graphs, where F and G are fixed graphs with F a partial subgraph of G. Given S, a collection of G-subgraphs of graph K, an F - G cover of S is a set of T of F-subgraphs of K such that each subgraph in S contains as a subgraph a member of T. An F - G packing of S is a subcollection S′? S such that no two subgraphs in S′ have an F-subgraph in common. K is F - G perfect if for all such S, the minimum cardinality of an F - G cover of S equals the maximum cardinality of an F - G packing of S. Thus K_i-perfect graphs are precisely K_i-1 - K_i perfect graphs. We develop a hypergraph characterization of F - G perfect graphs that leads to an alternate proof of previous results on K_i-perfect graphs as well as to a characterization of F - G perfect graphs for other instances of F and G.