5,314 research outputs found
Strong edge-colouring of sparse planar graphs
A strong edge-colouring of a graph is a proper edge-colouring where each
colour class induces a matching. It is known that every planar graph with
maximum degree has a strong edge-colouring with at most
colours. We show that colours suffice if the graph has girth 6, and
colours suffice if or the girth is at least 5. In the
last part of the paper, we raise some questions related to a long-standing
conjecture of Vizing on proper edge-colouring of planar graphs
No finite -regular matchstick graph exists
A graph is called a unit-distance graph in the plane if there is an
injective embedding of in the plane such that every pair of adjacent
vertices are at unit distance apart. If additionally the corresponding edges
are non-crossing and all vertices have the same degree we talk of a regular
matchstick graph. Due to Euler's polyhedron formula we have . The
smallest known -regular matchstick graph is the so called Harborth graph
consisting of vertices. In this article we prove that no finite
-regular matchstick graph exists.Comment: 15 pages, 12 figures, 2 table
Acyclic 5-Choosability of Planar Graphs Without Adjacent Short Cycles
The conjecture claiming that every planar graph is acyclic 5-choosable[Borodin et al., 2002] has been verified for several restricted classes of planargraphs. Recently, O. V. Borodin and A. O. Ivanova, [Journal of Graph Theory,68(2), October 2011, 169-176], have shown that a planar graph is acyclically 5-choosable if it does not contain an i-cycle adjacent to a j-cycle, where 3<=j<=5 if i=3 and 4<=j<=6 if i=4. We improve the above mentioned result and prove that every planar graph without an i-cycle adjacent to a j-cycle with3<=j<=5 if i=3 and 4<=j<=5 if i=4 is acyclically 5-choosable
Odd-Sum Colorings of Planar Graphs
A \emph{coloring} of a graph is a map such that
for all . A coloring is an \emph{odd-sum}
coloring if is odd, for each vertex . The
\emph{odd-sum chromatic number} of a graph , denoted , is the
minimum number of colors used (that is, the minimum size of the range) in an
odd-sum coloring of . Caro, Petru\v{s}evski, and \v{S}krekovski showed,
among other results, that is well-defined for every finite graph
and, in fact, . Thus, for
every planar graph (by the 4 Color Theorem), for every
triangle-free planar graph (by Gr\"{o}tzsch's Theorem), and
for every bipartite graph.
Caro et al. asked, for every even , whether there exists
such that if is planar with maximum degree and girth
at least then . They also asked, for every even
, whether there exists such that if is planar and
bipartite with maximum degree and girth at least then
. We answer both questions negatively. We also refute a
conjecture they made, resolve one further problem they posed, and make progress
on another.Comment: 8 pages, 6 figures, to appear in Discrete Applied Mat
On almost hypohamiltonian graphs
A graph is almost hypohamiltonian (a.h.) if is non-hamiltonian, there
exists a vertex in such that is non-hamiltonian, and is
hamiltonian for every vertex in . The second author asked in [J.
Graph Theory 79 (2015) 63--81] for all orders for which a.h. graphs exist. Here
we solve this problem. To this end, we present a specialised algorithm which
generates complete sets of a.h. graphs for various orders. Furthermore, we show
that the smallest cubic a.h. graphs have order 26. We provide a lower bound for
the order of the smallest planar a.h. graph and improve the upper bound for the
order of the smallest planar a.h. graph containing a cubic vertex. We also
determine the smallest planar a.h. graphs of girth 5, both in the general and
cubic case. Finally, we extend a result of Steffen on snarks and improve two
bounds on longest paths and longest cycles in polyhedral graphs due to
Jooyandeh, McKay, {\"O}sterg{\aa}rd, Pettersson, and the second author.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1602.0717
Graphs with few Hamiltonian Cycles
We describe an algorithm for the exhaustive generation of non-isomorphic
graphs with a given number of hamiltonian cycles, which is especially
efficient for small . Our main findings, combining applications of this
algorithm and existing algorithms with new theoretical results, revolve around
graphs containing exactly one hamiltonian cycle (1H) or exactly three
hamiltonian cycles (3H). Motivated by a classic result of Smith and recent work
of Royle, we show that there exist nearly cubic 1H graphs of order iff is even. This gives the strongest form of a theorem of Entringer and
Swart, and sheds light on a question of Fleischner originally settled by
Seamone. We prove equivalent formulations of the conjecture of Bondy and
Jackson that every planar 1H graph contains two vertices of degree 2, verify it
up to order 16, and show that its toric analogue does not hold. We treat
Thomassen's conjecture that every hamiltonian graph of minimum degree at least
contains an edge such that both its removal and its contraction yield
hamiltonian graphs. We also verify up to order 21 the conjecture of Sheehan
that there is no 4-regular 1H graph. Extending work of Schwenk, we describe all
orders for which cubic 3H triangle-free graphs exist. We verify up to order
Cantoni's conjecture that every planar cubic 3H graph contains a triangle,
and show that there exist infinitely many planar cyclically 4-edge-connected
cubic graphs with exactly four hamiltonian cycles, thereby answering a question
of Chia and Thomassen. Finally, complementing work of Sheehan on 1H graphs of
maximum size, we determine the maximum size of graphs containing exactly one
hamiltonian path and give, for every order , the exact number of such graphs
on vertices and of maximum size.Comment: 29 pages; to appear in Mathematics of Computatio
Circuits and Cycles in Graphs and Matroids
This dissertation mainly focuses on characterizing cycles and circuits in graphs, line graphs and matroids. We obtain the following advances.
1. Results in graphs and line graphs. For a connected graph G not isomorphic to a path, a cycle or a K1,3, let pc(G) denote the smallest integer n such that the nth iterated line graph Ln(G) is panconnected. A path P is a divalent path of G if the internal vertices of P are of degree 2 in G. If every edge of P is a cut edge of G, then P is a bridge divalent path of G; if the two ends of P are of degree s and t, respectively, then P is called a divalent (s, t)-path. Let l(G) = max{m : G has a divalent path of length m that is not both of length 2 and in a K3}. We prove the following. (i) If G is a connected triangular graph, then L(G) is panconnected if and only if G is essentially 3-edge-connected. (ii) pc(G) ≤ l(G) + 2. Furthermore, if l(G) ≥ 2, then pc(G) = l(G) + 2 if and only if for some integer t ≥ 3, G has a bridge divalent (3, t)-path of length l(G).
For a graph G, the supereulerian width μ′(G) of a graph G is the largest integer s such
that G has a spanning (k;u,v)-trail-system, for any integer k with 1 ≤ k ≤ s, and for any
u, v ∈ V (G) with u ̸= v. Thus μ′(G) ≥ 2 implies that G is supereulerian, and so graphs with
higher supereulerian width are natural generalizations of supereulerian graphs. Settling an open
problem of Bauer, Catlin in [J. Graph Theory 12 (1988), 29-45] proved that if a simple graph
G on n ≥ 17 vertices satisfy δ(G) ≥ n − 1, then μ′(G) ≥ 2. In this paper, we show that for 4
any real numbers a, b with 0 \u3c a \u3c 1 and any integer s \u3e 0, there exists a finite graph family
F = F(a,b,s) such that for a simple graph G with n = |V(G)|, if for any u,v ∈ V(G) with
uv ∈/ E(G), max{dG(u), dG(v)} ≥ an + b, then either μ′(G) ≥ s + 1 or G is contractible to a
member in F. When a = 1,b = −3, we show that if n is sufficiently large, K3,3 is the only 42
obstacle for a 3-edge-connected graph G to satisfy μ′(G) ≥ 3. An hourglass is a graph obtained from K5 by deleting the edges in a cycle of length 4, and an
hourglass-free graph is one that has no induced subgraph isomorphic to an hourglass. Kriesell in [J. Combin. Theory Ser. B, 82 (2001), 306-315] proved that every 4-connected hourglass-free line graph is Hamilton-connected, and Kaiser, Ryj ́aˇcek and Vr ́ana in [Discrete Mathematics, 321 (2014) 1-11] extended it by showing that every 4-connected hourglass-free line graph is 1- Hamilton-connected. We characterize all essentially 4-edge-connected graphs whose line graph is hourglass-free. Consequently we prove that for any integer s and for any hourglass-free line
graph L(G), each of the following holds. (i) If s ≥ 2, then L(G) is s-hamiltonian if and only if κ(L(G)) ≥ s + 2; (ii) If s ≥ 1, then L(G) is s-Hamilton-connected if and only if κ(L(G)) ≥ s + 3.
For integers s1, s2, s3 \u3e 0, let Ns1,s2,s3 denote the graph obtained by identifying each vertex of a K3 with an end vertex of three disjoint paths Ps1+1, Ps2+1, Ps3+1 of length s1,s2 and s3, respectively. We prove the following results. (i)LetN1 ={Ns1,s2,s3 :s1 \u3e0,s1 ≥s2 ≥s3 ≥0ands1+s2+s3 ≤6}. Thenforany N ∈ N1, every N-free line graph L(G) with |V (L(G))| ≥ s + 3 is s-hamiltonian if and only if κ(L(G)) ≥ s + 2. (ii)LetN2={Ns1,s2,s3 :s1\u3e0,s1≥s2≥s3≥0ands1+s2+s3≤4}.ThenforanyN∈N2, every N -free line graph L(G) with |V (L(G))| ≥ s + 3 is s-Hamilton-connected if and only if κ(L(G)) ≥ s + 3. 2. Results in matroids. A matroid M with a distinguished element e0 ∈ E(M) is a rooted matroid with e0 being the root. We present a characterization of all connected binary rooted matroids whose root lies in at most three circuits, and a characterization of all connected binary rooted matroids whose root lies in all but at most three circuits. While there exist infinitely many such matroids, the number of serial reductions of such matroids is finite. In particular, we find two finite families of binary matroids M1 and M2 and prove the following. (i) For some e0 ∈ E(M), M has at most three circuits containing e0 if and only if the serial reduction of M is isomorphic to a member in M1. (ii) If for some e0 ∈ E(M), M has at most three circuits not containing e0 if and only if the serial reduction of M is isomorphic to a member in M2. These characterizations will be applied to show that every connected binary matroid M with at least four circuits has a 1-hamiltonian circuit graph
Combinatorial Properties of Triangle-Free Rectangle Arrangements and the Squarability Problem
We consider arrangements of axis-aligned rectangles in the plane. A geometric
arrangement specifies the coordinates of all rectangles, while a combinatorial
arrangement specifies only the respective intersection type in which each pair
of rectangles intersects. First, we investigate combinatorial contact
arrangements, i.e., arrangements of interior-disjoint rectangles, with a
triangle-free intersection graph. We show that such rectangle arrangements are
in bijection with the 4-orientations of an underlying planar multigraph and
prove that there is a corresponding geometric rectangle contact arrangement.
Moreover, we prove that every triangle-free planar graph is the contact graph
of such an arrangement. Secondly, we introduce the question whether a given
rectangle arrangement has a combinatorially equivalent square arrangement. In
addition to some necessary conditions and counterexamples, we show that
rectangle arrangements pierced by a horizontal line are squarable under certain
sufficient conditions.Comment: 15 pages, 13 figures, extended version of a paper to appear at the
International Symposium on Graph Drawing and Network Visualization (GD) 201
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