Long Cycles in 2-Connected Triangle-Free Graphs

Graphs and Combinatorics

A set S of vertices in a graph G without isolated vertices is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ t (G) of G. The girth of G is the length of a shortest cycle in G. Let G be a connected graph with minimum degree at least 2, order n and girth g ≥ 3. It was shown in an earlier manuscript (Henning and Yeo in Graphs Combin 24:333–348, 2008) that $${\gamma_t(G)\le(\frac{1}{2}+\frac{1}{g})n}$$, and this bound is sharp for cycles of length congruent to two modulo four. In this paper we show that $${\gamma_t(G)\le\frac{n}{2}+\max(1,\frac{n}{2(g+1)})}$$, and this bound is sharp.

Discrete Applied Mathematics, 2009

We prove sharp bounds concerning domination number, radius, order and minimum degree of a graph. In particular, we prove that if G is a connected graph of order n, domination number γ and radius r, then 2 3 r ≤ γ ≤ n − 4 3 r + 2 3 . Equality is achieved in the upper bound if, and only if, G is a path or a cycle on n vertices with n ≡ 4(mod 6). Further, if G has minimum degree δ ≥ 3 and r ≥ 6, then using a result due to Erdös, Pach, Pollack, and Tuza [P. Erdös, J. Pach, R. Pollack, Z. Tuza, Radius, diameter, and minimum degree. J.

arXiv (Cornell University), 2015

The k-dominating graph D k (G) of a graph G is defined on the vertex set consisting of dominating sets of G with cardinality at most k, two such sets being adjacent if they differ by either adding or deleting a single vertex. A graph is a dominating graph if it is isomorphic to D k (G) for some graph G and some positive integer k. Answering a question of Haas and Seyffarth for graphs without isolates, it is proved that if G is such a graph of order n ≥ 2 and with G ∼ = D k (G), then k = 2 and G = K 1,n-1 for some n ≥ 4. It is also proved that for a given r there exist only a finite number of r-regular, connected dominating graphs of connected graphs. In particular, C 6 and C 8 are the only dominating graphs in the class of cycles. Some results on the order of dominating graphs are also obtained.

Combinatorica, 1998

To the memory of Paul Erdős Assume that G is a triangle-free graph. Let h d (G) be the minimum number of edges one has to add to G to get a graph of diameter at most d which is still triangle-free. It is shown that h 2 (G)= Θ(n log n) for connected graphs of order n and of fixed maximum degree. The proof is based on relations of h 2 (G) and the clique-cover number of edges of graphs. It is also shown that the maximum value of h 2 (G) over (triangle-free) graphs of order n is n/2 − 1 n/2 − 1. The behavior of h 3 (G) is different, its maximum value is n − 1. We could not decide whether h 4 (G) ≤ (1 −)n for connected (triangle-free) graphs of order n with a positive .

Ars Combinatoria Waterloo Then Winnipeg, 2006

MacGillivray and Seyffarth (J. Graph Theory 22 (1996), 213-229) proved that planar graphs of diameter two have domination number at most three and planar graphs of diameter three have domination number at most ten. They also give examples of planar graphs of diameter four having arbitrarily large domination numbers. In this paper we improve on their results. We prove that there is in fact a unique planar graph of diameter two with domination number three, and all other planar graphs of diameter two have domination number at most two. We also prove that every planar graph of diameter three and of radius two has domination number at most six. We then show that every sufficiently large planar graph of diameter three has domination number at most seven. Analogous results for other surfaces are discussed.

For the class of triangle-free graphs Brooks' Theorem can be restated in terms of forbidden induced subgraphs, i.e. let G be a triangle-free and K 1,r+1-free graph. Then G is r-colourable unless G is isomorphic to an odd cycle or a complete graph with at most two vertices. In this note we present an improvement of Brooks' Theorem for triangle-free and rsunshade-free graphs. Here, an r-sunshade (with r ≥ 3) is a star K 1,r with one branch subdivided. A classical result in graph colouring theory is the theorem of Brooks [2], asserting that every graph G is (∆(G))-colourable unless G is isomorphic to an odd cycle or a complete graph. Bryant [3] simplified this proof with the following characterization of cycles and complete graphs. Thereby he highlights the exceptional role of the cycles and complete graphs in Brooks' Theorem. Here we give a new elementary proof of this characterization. Proposition 1 (Bryant [3]). Let G be a 2-connected graph. Then G is a cycle or a complete graph if and only if G − {u, v} is not connected for every pair (u, v) of vertices of distance two. Proof. Let G be a 2-connected graph of order n. If G is a cycle or a complete graph, then obviously G − {u, v} is not connected for every pair (u, v) of vertices of distance two. Hence, assume that G is neither a cycle nor a complete graph and that G − {u, v} is not connected for every pair (u, v) of vertices of distance two. Note that then there exists at least one vertex v of G with 2 < d G (v) < n − 1. Since G

Journal of Graph Theory, 2013

The total domination number γ t (G) of a graph G is the minimum cardinality of a set S of vertices, so that every vertex of G is adjacent to a vertex in S. In this article, we determine an optimal upper bound on the total domination number of a graph with diameter 2. We show that for every graph G on n vertices with diameter 2, γ t (G) ≤ 1 + n ln(n). This bound is optimal in the sense that given any > 0, there exist graphs G with diameter 2 of all sufficiently large even orders n such that γ t (G) > (1 4+) n ln(n).

Graphs and Combinatorics, 2002

Let Gn; 3; 5;. .. ; 2k 1 denote the class of non-bipartite graphs on n vertices having no odd cycle of length 2k 1. We prove that eG 1 4 n À 2k 1 2 j k 2k À 1 for every G P Gn; 3; 5;. .. ; 2k 1 and characterize the extremal graphs. We also study the subclass Hn; 3; 5;. .. ; 2k 1 consisting of the hamiltonian members of Gn; 3; 5;. .. ; 2k 1. For this subclass the above upper bound holds for odd n. For even n we establish the following sharp upper bound: eG 1 4 nn À 8k 8 4k 2 À 7; and characterize the extremal graphs.

Graphs and Combinatorics, 2001

Let G be a graph of order n with connectivity j ! 3 and let a be the independence number of G. Set r 4 G minf P 4 i1 dx i : fx 1 ; x 2 ; x 3 ; x 4 g is an independent set of Gg. In this paper, we will prove that if r 4 G ! n 2j, then there exists a longest cycle C of G such that V G À C is an independent set of G. Furthermore, if the minimum degree of G is at least a, then G is hamiltonian.

Journal of Graph Theory, 2002

MacGillivray and Seyffarth (J. Graph Theory 22 (1996), 213-229) proved that planar graphs of diameter two have domination number at most three and planar graphs of diameter three have domination number at most ten. They also give examples of planar graphs of diameter four having arbitrarily large domination numbers. In this paper we improve on their results. We prove that there is in fact a unique planar graph of diameter two with domination number three, and all other planar graphs of diameter two have domination number at most two. We also prove that every planar graph of diameter three and of radius two has domination number at most six. We then show that every sufficiently large planar graph of diameter three has domination number at most seven. Analogous results for other surfaces are discussed.