###### On Uniquely Hamiltonian Claw-Free and Triangle-Free Graphs

## Abstract

A graph is uniquely Hamiltonian if it contains exactly one Hamiltonian cycle. In this note, we prove that claw-free graphs with minimum degree at least 3 are not uniquely Hamiltonian. We also show that this is best possible by exhibiting uniquely Hamiltonian claw-free graphs with minimum degree 2 and arbitrary maximum degree. Finally, we show that a construction due to Entringer and Swart can be modified to construct triangle-free uniquely Hamiltonian graphs with minimum degree 3.

###### Simple Graphs as Simplicial Complexes: the Mycielskian of a Graph

## Summary

Harary [10, p. 7] claims that Veblen [20, p. 2] first suggested to formalize simple graphs using simplicial complexes. We have developed basic terminology for simple graphs as at most 1-dimensional complexes.

We formalize this new setting and then reprove Mycielski’s [12] construction resulting in a triangle-free graph with arbitrarily large chromatic number. A different formalization of similar material is in [15].

###### Domination, Eternal Domination, and Clique Covering

## Abstract

Eternal and m-eternal domination are concerned with using mobile guards to protect a graph against infinite sequences of attacks at vertices. Eternal domination allows one guard to move per attack, whereas more than one guard may move per attack in the m-eternal domination model. Inequality chains consisting of the domination, eternal domination, m-eternal domination, independence, and clique covering numbers of graph are explored in this paper.

Among other results, we characterize bipartite and triangle-free graphs with domination and eternal domination numbers equal to two, trees with equal m-eternal domination and clique covering numbers, and two classes of graphs with equal domination, eternal domination and clique covering numbers.

###### Sufficient Conditions for Maximally Edge-Connected and Super-Edge-Connected Graphs Depending on The Clique Number

## Abstract

Let *G* be a connected graph with minimum degree *δ* and edge-connectivity *λ*. A graph is maximally edge-connected if *λ* = *δ*, and it is super-edgeconnected if every minimum edge-cut is trivial; that is, if every minimum edge-cut consists of edges incident with a vertex of minimum degree. The clique number *ω*(*G*) of a graph *G* is the maximum cardinality of a complete subgraph of *G*. In this paper, we show that a connected graph *G* with clique number *ω*(*G*) ≤ *r* is maximally edge-connected or super-edge-connected if the number of edges is large enough. These are generalizations of corresponding results for triangle-free graphs by Volkmann and Hong in 2017.

###### On the Non-(p−1)-Partite Kp-Free Graphs

References [1] N. Alon, P. Erdős, R. Holzman and M. Krivelevich, On k-saturated graphs with restrictions on the degrees, J. Graph Theory 23(1996) 1-20. doi:10.1002/(SICI)1097-0118(199609)23:1h1::AID-JGT1i3.0.CO;2-O [2] K. Amin, J.R. Faudree and R.J. Gould, The edge spectrum of K4-saturated graphs, J. Combin. Math. Combin. Comp. 81 (2012) 233-242. [3] C. Barefoot, K. Casey, D. Fisher, K. Fraughnaugh and F. Harary, Size in maximal triangle-free graphs and minimal graphs of diameter 2, Discrete Math. 138

###### Making a Dominating Set of a Graph Connected

## Abstract

Let *G* = (*V,E*) be a graph and *S* ⊆ *V*. We say that *S* is a dominating set of *G*, if each vertex in *V* \ *S* has a neighbor in *S*. Moreover, we say that *S* is a connected (respectively, 2-edge connected or 2-connected) dominating set of *G* if *G*[*S*] is connected (respectively, 2-edge connected or 2-connected). The domination (respectively, connected domination, or 2- edge connected domination, or 2-connected domination) number of *G* is the cardinality of a minimum dominating (respectively, connected dominating, or 2-edge connected dominating, or 2-connected dominating) set of *G*, and is denoted γ(*G*) (respectively γ_{1}(*G*), or γ′ _{2}(*G*), or γ_{2}(*G*)). A well-known result of Duchet and Meyniel states that γ_{1}(*G*) ≤ 3γ(*G*) − 2 for any connected graph *G*. We show that if γ(*G*) ≥ 2, then γ′ _{2}(*G*) ≤ 5γ(*G*) − 4 when *G* is a 2-edge connected graph and γ_{2}(*G*) ≤ 11γ(*G*) − 13 when *G* is a 2-connected triangle-free graph.

###### Saturation Spectrum of Paths and Stars

R eferences [1] K. Amin, J. Faudree and R.J. Gould, The edge spectrum of K 4 -saturated graphs , J. Combin. Math. Combin. Comput. 81 (2012) 233–242. [2] K. Amin, J. Faudree, R.J. Gould and E. Sidorowicz, On the non- ( p − 1) -partite K p -free graphs , Discuss. Math. Graph Theory 33 (2013) 9–23. doi:10.7151/dmgt.1654 [3] C. Barefoot, K. Casey, D. Fisher, K. Fraughnaugh and F. Harary, Size in maximal triangle-free graphs and minimal graphs of diameter 2, Discrete Math. 138 (1995) 93–99. doi:10.1016/0012-365X(94)00190-T [4] G

###### Large Degree Vertices in Longest Cycles of Graphs, I

in graphs with prescribed stability number and connectivity, J. Combin. Theory Ser. B 60 (1994) 315-318. doi:10.1006/jctb.1994.1023 [6] B. Li and S. Zhang, Forbidden subgraphs for longest cycles to contain vertices with large degree, Discrete Math. 338 (2015) 1681-1689. doi:10.1016/j.disc.2014.07.003 [7] D. Paulusma and K. Yoshimoto, Cycles through specified vertices in triangle-free graphs, Discuss. Math. Graph Theory 27 (2007) 179-191. doi:10.7151/dmgt.1354 [8] A. Saito, Long cycles through specified vertices in a

###### Chromatic Properties of the Pancake Graphs

of a graph , Discrete Math. 24 (1978) 1–6. doi:10.1016/0012-365X(78)90167-X [5] I.J. Dejter and O. Serra, Efficient dominating sets in Cayley graphs , Discrete Appl. Math. 129 (2003) 319–328. doi:10.1016/S0166-218X(02)00573-5 [6] H. Dweighter, E 2569 in: Elementary problems and solutions , Amer. Math. Monthly 82 (1975) 1010. [7] A. Johansson, Asymptotic choice number for triangle free graphs, DIMACS Technical Report (1996) 91–95. [8] A. Kanevsky and C. Feng, On the embedding of cycles in Pancake graphs , Parallel Comput. 21

###### On Independent Domination in Planar Cubic Graphs

] W. Goddard, M. A. Henning, J. Lyle and J. Southey, On the independent domination number of regular graphs , Ann. Comb. 16 (2012) 719–732. doi:10.1007/s00026-012-0155-4 [5] W. Goddard and J. Lyle, Independent dominating sets in triangle-free graphs , J. Comb. Optim. 23 (2012) 9–20. doi:10.1007/s10878-010-9336-4 [6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc. New York, 1998). [7] M.A. Henning, C. Löwenstein and D. Rautenbach, Independent domination in sub- cubic bipartite graphs of