Vertices Of Odd Degree Research Articles

A (Slightly) Improved Approximation Algorithm for Metric TSP

In “An Improved Approximation Algorithm for TSP,” Karlin, Klein, and Oveis Gharan design the first improvement over the classical 1.5 approximation algorithm of Christofides-Serdyukov after more than 40 years. Their algorithm first chooses a random spanning tree from the maximum entropy distribution of spanning trees with marginals equal to the optimum LP solution of TSP, and then, similar to Christofides’ algorithm, it adds the minimum cost matching on the odd degree vertices of the tree. To analyze their simple algorithms, they prove and exploit new tools from the theory of strongly Rayleigh distributions.

Extremal Harary Index of Graphs with Given Number of Vertices of Odd Degree

The Harary index is one of the most investigated topological indices in mathematical chemistry and in related disciplines, which can be used to describe the molecular structure and physical properties. Let G n , 2 k be the set of all connected graphs on n vertices with exactly 2 k vertices of odd degree. Recently, the graphs in G n , 0 with the first four maximal Harary index and the first minimal Harary index are determined. In the paper, by using the edge-deleted method and some combinatorial skills, we characterize the graph in G n , 2 k with the maximal Harary index for all 0 ≤ k ≤ n / 2 , the graphs in G n , 2 with the first two minimal Harary index, and the graph in G n , 4 with the minimal Harary index. These results will be conducive to the further study of Harary index and its applications.

Equitable factorizations of edge-connected graphs

In this paper, we show that every ( 3 k − 3 ) -edge-connected graph G , under a certain degree condition, can be edge-decomposed into k factors G 1 , … , G k such that for each vertex v ∈ V ( G i ) , | d G i ( v ) − d G ( v ) / k | < 1 , where 1 ≤ i ≤ k . As an application, we deduce that every 6 -edge-connected graph G can be edge-decomposed into three factors G 1 , G 2 , and G 3 such that for each vertex v ∈ V ( G i ) with 1 ≤ i ≤ 3 , | d G i ( v ) − d G ( v ) / 3 | < 1 , unless G has exactly one vertex z with d G ( z ) ⁄ ≡ 3 0 . Next, we show that every odd- ( 3 k − 2 ) -edge-connected graph G can be edge-decomposed into k factors G 1 , … , G k such that for each vertex v ∈ V ( G i ) , d G i ( v ) and d G ( v ) have the same parity and | d G i ( v ) − d G ( v ) / k | < 2 , where k is an odd positive integer and 1 ≤ i ≤ k . Finally, we give a sufficient edge-connectivity condition for a graph G to have a parity factor F with specified odd-degree vertices such that for each vertex v , | d F ( v ) − ɛ d G ( v ) | < 2 , where ɛ is a real number with 0 < ɛ < 1 .

Extremal Wiener Index of Graphs with Given Number of Vertices of Odd Degree

Extremal Wiener Index of Graphs with Given Number of Vertices of Odd Degree

A parity theorem about trees with specified degrees

Carsten Thomassen and I proved that in any graph G, the number of cycles containing a specified edge as well as all the vertices of odd degree is odd if and only if G is eulerian. Where all vertices have even degree this is a theorem of Shunichi Toida and where all vertices have odd degree it is Andrew Thomason’s extension of Smith’s Theorem. Our proof was not algorithmic. In this paper, I extend Thomason’s algorithm to one which, in a non-eulerian graph, finds a second cycle containing a specified edge and all the odd-degree vertices.Kenneth Berman extended Thomason’s Theorem to spanning trees: he proved that if T is a spanning tree such that the degree of every vertex in G−E(T) is odd, then there is an even number of spanning trees of G with the same degree as T at each vertex.In this paper, I give a common generalization of all of these theorems to a parity theorem about “good” trees in general graphs. My proof provides an algorithm for finding a second “good” tree.

Path and cycle decompositions of dense graphs

We make progress on three long standing conjectures from the 1960s about path and cycle decompositions of graphs. Gallai conjectured that any connected graph on $n$ vertices can be decomposed into at most $\left\lceil \frac{n}{2}\right\rceil$ paths, while a conjecture of Haj\'{o}s states that any Eulerian graph on $n$ vertices can be decomposed into at most $\left\lfloor \frac{n-1}{2}\right\rfloor$ cycles. The Erd\H{o}s-Gallai conjecture states that any graph on $n$ vertices can be decomposed into $O(n)$ cycles and edges. We show that if $G$ is a sufficiently large graph on $n$ vertices with linear minimum degree, then the following hold. (i) $G$ can be decomposed into at most $\frac{n}{2}+o(n)$ paths. (ii) If $G$ is Eulerian, then it can be decomposed into at most $\frac{n}{2}+o(n)$ cycles. (iii) $G$ can be decomposed into at most $\frac{3 n}{2}+o(n)$ cycles and edges. If in addition $G$ satisfies a weak expansion property, we asymptotically determine the required number of paths/cycles for each such $G$. (iv) $G$ can be decomposed into $\max \left\{\frac{odd(G)}{2},\frac{\Delta(G)}{2}\right\}+o(n)$ paths, where $odd(G)$ is the number of odd-degree vertices of $G$. (v) If $G$ is Eulerian, then it can be decomposed into $\frac{\Delta(G)}{2}+o(n)$ cycles. All bounds in (i)-(v) are asymptotically best possible.

The complexity of the parity argument with potential

The parity argument principle states that every finite graph has an even number of odd degree vertices. We consider the problem whose totality is guaranteed by the parity argument on a graph with potential. In this paper, we show that the problem of finding an unknown odd-degree vertex or a local optimum vertex on a graph with potential is polynomially equivalent to EndOfPotentialLine if the maximum degree is at most three. However, even if the maximum degree is 4, such a problem is PPA∩PLS-complete. To show the complexity of this problem, we provide new results on multiple-source variants of EndOfPotentialLine, which is the canonical problem for EOPL. This result extends the work by Goldberg and Hollender; they studied similar variants of EndOfLine.

Towards Gallai's path decomposition conjecture

AbstractA path decomposition of a graph is a collection of edge‐disjoint paths of that covers the edge set of . Gallai conjectured that every connected graph on vertices admits a path decomposition of cardinality at most . Seminal results toward its verification consider the graph obtained from by removing its vertices of odd degree, which is called the E‐subgraph of . Lovász verified Gallai's Conjecture for graphs whose E‐subgraphs consist of at most one vertex, and Pyber verified it for graphs whose E‐subgraphs are forests. In 2005, Fan verified Gallai's Conjecture for graphs in which each block, that is, each maximal 2‐connected subgraph, of their E‐subgraph is triangle‐free and has maximum degree at most 3. Let be the family of graphs for which (a) each block has maximum degree at most ; and (b) each component either has maximum degree at most 3 or has at most one block that contains triangles. In this paper, we generalize Fan's result by verifying Gallai's Conjecture for graphs whose E‐subgraphs are subgraphs of graphs in . This allows the components of the E‐subgraphs to contain any number of blocks with triangles as long as they are subgraphs of graphs in .

N-Flips in Triangulations with Two Odd Degree Vertices

In this paper, we prove that any two triangulations G and $$G'$$ on the sphere with exactly two odd degree vertices can be transformed into each other by two local transformations, called an N-flip and a $$P_2$$-flip, preserving the parity of degree of each vertex, if $$|V(G)|=|V(G')|$$. This is an analogy of the same result for triangulations with each vertex even degree [7], but we prove such a fact does not hold for triangulations with at least four odd degree vertices.

Cycles containing all the odd-degree vertices

The number of cycles in a graph containing any fixed edge and also containing all vertices of odd degree is odd if and only if all vertices have even degree. If all vertices have even degree this is a theorem of Shunichi Toida. If all vertices have odd degree it is Andrew Thomason's extension of Smith's theorem.

New classes of set-sequential trees

A graph is called set-sequential if its vertices can be labeled with distinct nonzero vectors in F2n such that when each edge is labeled with the sum(mod2) of its vertices, every nonzero vector in F2n is the label for either a single vertex or a single edge. We resolve certain cases of a conjecture of Balister, Győri, and Schelp in order to show many new classes of trees to be set-sequential. We show that all caterpillars T of diameter k such that k≤18 or |V(T)|≥2k−1 are set-sequential, where T has only odd-degree vertices and |V(T)|=2n−1 for some positive integer n. We also present a new method of recursively constructing set-sequential trees.

Gallai's path decomposition conjecture for graphs with treewidth at most 3

AbstractA path decomposition of a graph is a set of edge‐disjoint paths of that covers the edge set of . Gallai (1968) conjectured that every connected graph with vertices admits a path decomposition of size at most . Gallai's conjecture was verified for many classes of graphs. In particular, Lovász (1968) verified this conjecture for graphs with at most one vertex of even degree, and Pyber (1996) verified it for graphs in which every cycle contains a vertex of odd degree. Recently, Bonamy and Perrett verified Gallai's conjecture for graphs with maximum degree at most 5. In this paper, we verify Gallai's conjecture for graphs with treewidth at most 3. Moreover, we show that the only graphs with treewidth at most 3 that do not admit a path decomposition of size at most are isomorphic to or , the graph obtained from by removing an edge.

Gallai's conjecture for graphs with treewidth 3

Gallai conjectured (1966) that the edge-set of a simple graph G with n vertices can be covered by at most (n+1)/2 edge-disjoint paths. Lovász [Lovász, L., On covering of graphs, in: Theory of Graphs (Proc. Colloq., Tihany, 1966), Academic Press, New York, 1968 pp. 231–236.] verified this conjecture for graphs with at most one vertex of even degree, and Pyber [Pyber, L., Covering the edges of a connected graph by paths, J. Combin. Theory Ser. B 66 (1996), pp. 152–159.] verified it for graphs in which every cycle contains a vertex of odd degree. Recently, Bonamy and Perrett verified this Conjecture for graphs with maximum degree at most 5. In this paper, we verify the Conjecture for graphs with treewidth at most 3.

Even Cycles and Even 2-Factors in the Line Graph of a Simple Graph

Let $G$ be a connected graph with an even number of edges. We show that if the subgraph of $G$ induced by the vertices of odd degree has a perfect matching, then the line graph of $G$ has a $2$-factor whose connected components are cycles of even length (an even $2$-factor). For a cubic graph $G$, we also give a necessary and sufficient condition so that the corresponding line graph $L(G)$ has an even cycle decomposition of index $3$, i.e., the edge-set of $L(G)$ can be partitioned into three $2$-regular subgraphs whose connected components are cycles of even length. The more general problem of the existence of even cycle decompositions of index $m$ in $2d$-regular graphs is also addressed.

An Experimental Evaluation of the Best-of-Many Christofides’ Algorithm for the Traveling Salesman Problem

Recent papers on approximation algorithms for the traveling salesman problem (TSP) have given a new variant of the well-known Christofides' algorithm for the TSP, called the Best-of-Many Christofides' algorithm. The algorithm involves sampling a spanning tree from the solution to the standard LP relaxation of the TSP, subject to the condition that each edge is sampled with probability at most its value in the LP relaxation. One then runs Christofides' algorithm on the tree by computing a minimum-cost matching on the odd-degree vertices in the tree, and shortcutting a traversal of the resulting Eulerian graph to a tour. In this paper we perform an experimental evaluation of the Best-of-Many Christofides' algorithm to see if there are empirical reasons to believe its performance is better than that of Christofides' algorithm. Furthermore, several different sampling schemes have been proposed; we implement several different schemes to determine which ones might be the most promising for obtaining improved performance guarantees over that of Christofides' algorithm. In our experiments, all of the implemented methods perform significantly better than Christofides' algorithm; an algorithm that samples from a maximum entropy distribution over spanning trees seems to be particularly good, though there are others that perform almost as well.

Grünbaum colorings of triangulations on the projective plane

A Grünbaum coloring of a triangulation G on a surface is a 3-edge coloring of G such that each face of G receives three distinct colors on its boundary edges. In this paper, we prove that every Fisk triangulation on the projective plane P has a Grünbaum coloring, where a “Fisk triangulation” is one with exactly two odd degree vertices such that the two odd vertices are adjacent. To prove the theorem, we establish a generating theorem for Fisk triangulations on P. Moreover, we show that a triangulation G on P has a Grünbaum coloring with each color-induced subgraph connected if and only if every vertex of G has even degree.

A statistic related to trees and words on a finite alphabet

Let [Formula: see text] denote the set of trees on [Formula: see text] labeled vertices. In this paper, we consider the statistic on [Formula: see text] which records the number of vertices of odd degree and count trees having a prescribed number of such vertices. In doing so, we make use of a well-known bijection between members of [Formula: see text] and the set of [Formula: see text]-ary words of length [Formula: see text]. In fact, we are able to study a generalization mod [Formula: see text] of the aforementioned statistic for all [Formula: see text]. In the cases when [Formula: see text] and [Formula: see text], further enumerative results for both words and trees may be given.

Snarks, Hypohamiltonian Graphs and Non-Supereulerian Graphs

A graph G is hypohamiltonian if it is not Hamiltonian but for each $$v\in V(G)$$vźV(G), the graph $$G-v$$G-v is Hamiltonian. A graph is supereulerian if it has a spanning Eulerian subgraph. A graph G is called collapsible if for every even subset $$R\subseteq V(G)$$R⊆V(G), there is a spanning connected subgraph H of G such that R is the set of vertices of odd degree in H. A graph is reduced if it has no nontrivial collapsible subgraphs. In this note, we first prove that all hypohamiltonian cubic graphs are reduced non-supereulerian graphs. Then we introduce an operation to construct graphs from hypohamiltonian cubic graphs such that the resulting graphs are 3-edge-connected non-supereulerian reduced graphs and cannot be contracted to a snark. This disproves two conjectures, one of which was first posed by Catlin et al. in [Congr. Num. 76:173---181, 1990] and in [J. Combin. Theory, Ser B 66:123---139, 1996], and was posed again by Li et al. in [Acta Math. Sin. English Ser 30(2):291---304, 2014] and by Yang in [Supereulerian graphs, hamiltonicity of graphs and several extremal problems in graphs, Ph. D. Dissertation, Universite Paris-Sub, September 27, 2013], respectively, the other one was posed by Yang 2013.

Optimal path and cycle decompositions of dense quasirandom graphs

Motivated by longstanding conjectures regarding decompositions of graphs into paths and cycles, we prove the following optimal decomposition results for random graphs. Let 0<p<1 be constant and let G∼Gn,p. Let odd(G) be the number of odd degree vertices in G. Then a.a.s. the following hold:(i)G can be decomposed into ⌊Δ(G)/2⌋ cycles and a matching of size odd(G)/2.(ii)G can be decomposed into max⁡{odd(G)/2,⌈Δ(G)/2⌉} paths.(iii)G can be decomposed into ⌈Δ(G)/2⌉ linear forests. Each of these bounds is best possible. We actually derive (i)–(iii) from ‘quasirandom’ versions of our results. In that context, we also determine the edge chromatic number of a given dense quasirandom graph of even order. For all these results, our main tool is a result on Hamilton decompositions of robust expanders by Kühn and Osthus.

Optimal path and cycle decompositions of dense quasirandom graphs

Motivated by longstanding conjectures regarding decompositions of graphs into paths and cycles, we prove the following optimal decomposition results for random graphs. Let 0<p<1 be constant and let G∼Gn,p. Let odd(G) be the number of odd degree vertices in G. Then a.a.s. the following hold:(i)G can be decomposed into ⌊Δ(G)/2⌋ cycles and a matching of size odd(G)/2.(ii)G can be decomposed into max⁡{odd(G)/2,⌈Δ(G)/2⌉} paths.(iii)G can be decomposed into ⌈Δ(G)/2⌉ linear forests. Each of these bounds is best possible. We actually derive (i)–(iii) from ‘quasi-random’ versions of our results. In that context, we also determine the edge chromatic number of a given dense quasirandom graph of even order. For all these results, our main tool is a result on Hamilton decompositions of robust expanders by Kühn and Osthus.