Diamond-free Graph Research Articles

A Sufficient Condition for Planar Graphs of Maximum Degree 6 to be Totally 7-Colorable

A total k-coloring of a graph is an assignment of k colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The total coloring conjecture (TCC) states that every simple graph G has a total ΔG+2-coloring, where ΔG is the maximum degree of G. This conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6. It is known that every planar graph G of ΔG≥9 or ΔG∈7,8 with some restrictions has a total ΔG+1-coloring. In particular, in (Shen and Wang, 2009), the authors proved that every planar graph with maximum degree 6 and without 4-cycles has a total 7-coloring. In this paper, we improve this result by showing that every diamond-free and house-free planar graph of maximum degree 6 is totally 7-colorable if every 6-vertex is not incident with two adjacent four cycles or three cycles of size p,q,ℓ for some p,q,ℓ∈3,4,4,3,3,4.

On Total Domination and Hop Domination in Diamond-Free Graphs

Two vertices in a graph are said to 2-step dominate each other if they are at distance 2 apart. A set S of vertices in a graph G is a hop dominating set of G if every vertex outside S is 2-step dominated by some vertex of S. The hop domination number, $$\gamma _{h}(G)$$, of G is the minimum cardinality of a hop dominating set of G. A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to at least one vertex of S. The total domination number, $$\gamma _{t}(G)$$, of G is the minimum cardinality of a total dominating set of G. It is known that if G is a triangle-free graph, then $$\gamma _{h}(G)\le \gamma _{t}(G)$$. But there are connected graphs G for which the difference $$\gamma _{h}(G)-\gamma _{t}(G)$$ can be made arbitrarily large. It would be interesting to find other classes of graphs G that satisfy $$\gamma _{h}(G)\le \gamma _{t}(G)$$. In this paper, we study the relationship between total domination number and hop domination number in diamond-free graph. We prove that if G is diamond-free graph of order n with the exception of two special graphs, then $$\gamma _{h}(G)- \gamma _{t}(G)\le \frac{n}{6}$$. Furthermore, we find two subclasses of diamond-free graphs G that satisfy $$\gamma _{h}(G)\le \gamma _{t}(G)$$ and generalize the known result.

On the Structure of (Even Hole, Kite)-Free Graphs

A hole is an induced cycle with at least four vertices. A hole is even if it has an even number of vertices. Even-hole-free graphs are being actively studied in the literature. It is not known whether even-hole-free graphs can be optimally colored in polynomial time. A diamond is the graph obtained from the clique of four vertices by removing one edge. A kite is a graph consists of a diamond and another vertex adjacent to a vertex of degree two in the diamond. Kloks et al. (J Combin Theory B 99:733–800, 2009) designed a polynomial-time algorithm to color an (even hole, diamond)-free graph. In this paper, we consider the class of (even hole, kite)-free graphs which generalizes (even hole, diamond)-free graphs. We prove that a connected (even hole, kite)-free graph is diamond-free, or the join of a clique and a diamond-free graph, or contains a clique cutset. This result, together with the result of Kloks et al., imply the existence of a polynomial time algorithm to color (even hole, kite)-free graphs. We also prove (even hole, kite)-free graphs are $$\beta $$ -perfect in the sense of Markossian, Gasparian and Reed. Finally, we establish the Vizing bound for (even hole, kite)-free graphs.

Powers of edge ideals with linear resolutions

ABSTRACTWe show that if G is a gap-free and diamond-free graph, then I(G)s has a linear minimal free resolution for every s≥2.

Clique-perfectness and balancedness of some graph classes

A graph is clique-perfect if the maximum size of a clique-independent set (a set of pairwise disjoint maximal cliques) and the minimum size of a clique-transversal set (a set of vertices meeting every maximal clique) coincide for each induced subgraph. A graph is balanced if its clique-matrix contains no square submatrix of odd size with exactly two ones per row and column. In this work, we give linear-time recognition algorithms and minimal forbidden induced subgraph characterizations of clique-perfectness and balancedness of P4-tidy graphs and a linear-time algorithm for computing a maximum clique-independent set and a minimum clique-transversal set for any P4-tidy graph. We also give a minimal forbidden induced subgraph characterization and a linear-time recognition algorithm for balancedness of paw-free graphs. Finally, we show that clique-perfectness of diamond-free graphs can be decided in polynomial time by showing that a diamond-free graph is clique-perfect if and only if it is balanced.

Mixed unit interval graphs

The class of intersection graphs of unit intervals of the real line whose ends may be open or closed is a strict superclass of the well-known class of unit interval graphs. We pose a conjecture concerning characterizations of such mixed unit interval graphs, verify parts of it in general, and prove it completely for diamond-free graphs. In particular, we characterize diamond-free mixed unit interval graphs by means of an infinite family of forbidden induced subgraphs, and we show that a diamond-free graph is mixed unit interval if and only if it has intersection representations using unit intervals such that all ends of the intervals are integral.

A characterization of diameter-2-critical graphs whose complements are diamond-free

A graph G is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. The complete graph on four vertices minus one edge is called a diamond, and a diamond-free graph has no induced diamond subgraph. In this paper we use an association with total domination to characterize the diameter-2-critical graphs whose complements are diamond-free. Murty and Simon conjectured that the number of edges in a diameter-2-critical graph G of order n is at most ⌊n2/4⌋ and that the extremal graphs are the complete bipartite graphs K⌊n/2⌋⌈n/2⌉. As a consequence of our characterization, we prove the Murty–Simon conjecture for graphs whose complements are diamond-free.

Unique intersectability of diamond-free graphs

For a graph G with vertices v 1 , v 2 , … , v n , a simple set representation of G is a family F = { S 1 , S 2 , … , S n } of distinct nonempty sets such that | S i ∩ S j | = 1 if v i v j is an edge in G , and | S i ∩ S j | = 0 otherwise. Let S ( F ) = ⋃ i = 1 n S i , and let ω s ( G ) denote the minimum | S ( F ) | of a simple set representation F of G . If, for every two minimum simple set representations F and F ′ of G , F can be obtained from F ′ by a bijective mapping from S ( F ′ ) to S ( F ) , then G is said to be s -uniquely intersectable. In this paper, we are concerned with the s -unique intersectability of diamond-free graphs, where a diamond is a K 4 with one edge deleted. Moreover, for a diamond-free graph G , we also derive a formula for computing ω s ( G ) .

On Winning Fast in Avoider-Enforcer Games

We analyze the duration of the unbiased Avoider-Enforcer game for three basic positional games. All the games are played on the edges of the complete graph on $n$ vertices, and Avoider's goal is to keep his graph outerplanar, diamond-free and $k$-degenerate, respectively. It is clear that all three games are Enforcer's wins, and our main interest lies in determining the largest number of moves Avoider can play before losing. Extremal graph theory offers a general upper bound for the number of Avoider's moves. As it turns out, for all three games we manage to obtain a lower bound that is just an additive constant away from that upper bound. In particular, we exhibit a strategy for Avoider to keep his graph outerplanar for at least $2n-8$ moves, being just $6$ short of the maximum possible. A diamond-free graph can have at most $d(n)=\lceil\frac{3n-4}{2}\rceil$ edges, and we prove that Avoider can play for at least $d(n)-3$ moves. Finally, if $k$ is small compared to $n$, we show that Avoider can keep his graph $k$-degenerate for as many as $e(n)$ moves, where $e(n)$ is the maximum number of edges a $k$-degenerate graph can have.

On the connectivity of [formula omitted]-diamond-free graphs

For a vertex v of a graph G , we denote by d ( v ) the degree of v . The local connectivity κ ( u , v ) of two vertices u and v in a graph G is the maximum number of internally disjoint u – v paths in G , and the connectivity of G is defined as κ ( G ) = min { κ ( u , v ) | u , v ∈ V ( G ) } . Clearly, κ ( u , v ) ≤ min { d ( u ) , d ( v ) } for all pairs u and v of vertices in G . Let δ ( G ) be the minimum degree of G . We call a graph G maximally connected when κ ( G ) = δ ( G ) and maximally local connected when κ ( u , v ) = min { d ( u ) , d ( v ) } for all pairs u and v of distinct vertices in G . For an integer p ≥ 2 , we define a p - diamond as the graph with p + 2 vertices, where two adjacent vertices have exactly p common neighbors, and the graph contains no further edges. For p = 2 , the 2-diamond is the usual diamond. We call a graph p - diamond-free if it contains no p -diamond as a (not necessarily induced) subgraph. In 2007, Dankelmann, Hellwig and Volkmann [P. Dankelmann, A. Hellwig, L. Volkmann, On the connectivity of diamond-free graphs, Discrete Appl. Math. 155 (2007), 2111–2117] proved that a connected diamond-free graph G of order n ≤ 3 δ ( G ) is maximally connected when δ ( G ) ≥ 3 . In this paper, we show that such graphs are even maximally local connected when n ≤ 3 δ ( G ) − 1 . Examples demonstrate that this bound is best possible. In addition, we present similar results for p -diamond-free graphs.