We want to study graphs, structurally, without looking at the labelling. 6. Erdos and Tuza conjectured that for any n-vertex K4-free graph G with ⌊n2/4⌋+1 edges, one can find at least (1+o(1))n216 K4-saturating edges. Standard theory on treewidth tells us that a graph of treewidth at most 2 is 2-degenerate (see http://en.wikipedia.org/wiki/Degeneracy_%28graph_theory%29 ), which means that all induced … Figure 1: The Wagner graph V8 Corollary 2.4 can be reinterpreted using the following convenient de nition. (Start with: how many edges must it have?) Series B", Journal of Combinatorial Theory. Draw, if possible, two different planar graphs with the same number of vertices, edges… GATE CS 2011 Graph Theory Discuss it. Planar Graph: A graph is said to be a planar graph if we can draw all its edges in the 2-D plane such that no two edges intersect each other. That is, the (a)Draw the isomorphism classes of connected graphs on 4 vertices, and give the vertex and edge An edge 2. Consider the graph G1 = G v, having 3 vertices and 4 edges, one vertex w having degree 2. We construct a graph with only 2n233 K4-saturating edges. A minor of a graph G is a graph obtained from G by contracting edges, deleting edges, and deleting isolated vertices; a proper minor of G is any minor other than G itself. Likewise, what is a k4 graph? A complete graph K4. A cycle is a closed walk which contains any edge at most one time. Mathematical Properties of Spanning Tree. Strong edge colouring of graphs was instructed by Fouquet and Jolivet . Dive into the research topics of 'On the number of K_{4}-saturating edges'. @article{f6f5e74ae967444bbb17d3450646cd2a. Observe that in general two vertices iand jof an oriented graph can be connected by two edges directed opposite to each other, i.e. (3 pts.) K3= Complete Graph of 4 Vertices K4 = Complete Graph of 4 Vertices 1) How many Hamiltonian circuits does it have? Furthermore, we prove that it is best possible, i.e., one can always find at least (1+o(1))2n233 K4-saturating edges in an n-vertex K4-free graph with ⌊n2/4⌋+1 edges.". Below are listed some of these invariants: The matrix is uniquely defined (note that it centralizes all permutations). 2 1) How many Hamiltonian circuits does it have? Spanning tree has n-1 edges, where n is the number of nodes (vertices). If the edges that exist in graph I are absent in another graph II, and if both graph I and graph II are combined together to form a complete graph, then graph I and graph II are called complements of each other. Let G2 = G1 w. Clearly, G2 has 2 vertices and 2 edges. we take the unlabelled graph) then these graphs are not the same. Let G2 = G1 w. Clearly, G2 has 2 vertices and 2 edges. A complete graph is a graph in which each pair of graph vertices is connected by an edge. Furthermore, we prove that it is best possible, i.e., one can always find at least (1+o(1))2n233 K4-saturating edges in an n-vertex K4-free graph with ⌊n2/4⌋+1 edges. Let G be a K4-free graph; an edge in its complement is a K4-saturating edge if the addition of this edge to G creates a copy of K4. How many vertices and how many edges do these graphs have? A graph Gis an ordered pair (V;E), where V is a nite set and graph, G E V 2 is a set of pairs of elements in V. The set V is called the set of vertices and Eis called the set of edges of G. vertex, edge The edge e= fu;vg2 Copyright: Copyright 2015 Elsevier B.V., All rights reserved.". Graph K4 is palanar graph, because it has a planar embedding as shown in. It is well-known that the $K_4$-minor-free graphs are exactly the graphs of treewidth at most two, see http://en.wikipedia.org/wiki/Forbidden_graph_characterization. By allowing V or E to be an inﬁnite set, we obtain inﬁnite graphs. In the following example, graph-I has two edges 'cd' and 'bd'. As an example, the left graph in Figure 1 has three vertices VG={v1,v2,v3}V_{G} = \{v_{1}, v_{2}, v_{3}\}VG… In this case, any path visiting all edges must visit some edges more than once. 1 Preliminaries De nition 1.1. 5. 30 When a connected graph can be drawn without any edges crossing, it is called planar.When a planar graph is drawn in this way, it divides the plane into regions called faces.. UR - http://www.scopus.com/inward/record.url?scp=84908176935&partnerID=8YFLogxK, UR - http://www.scopus.com/inward/citedby.url?scp=84908176935&partnerID=8YFLogxK, JO - Journal of Combinatorial Theory. Here is an example of a bipartite graph (left), and an example of a graph that is not bipartite. Line Graphs Math 381 | Spring 2011 Since edges are so important to a graph, sometimes we want to know how much of the graph is determined by its edges. Erdos and Tuza conjectured that for any n-vertex K4-free graph G with ⌊n2/4⌋+1 edges, one can find at least (1+o(1))n216 K4-saturating edges. If Gis the complete graph on nvertices, then ˜(K n) = nand n 2 is the number of edges … The one we’ll talk about is this: You know the edge … The graph K4 has six edges. A complete graph with n nodes represents the edges of an (n − 1)-simplex. the spanning tree is minimally connected. title = "On the number of K4-saturating edges". English: Complete bipartite graph K4,4 with colors showing edges from red vertices to blue vertices in green We construct a graph with only 2n233 K4-saturating edges. Draw, if possible, two different planar graphs with the same number of vertices, edges… A closed walk is a sequence of alternating vertices and edges that starts and ends at the same vertex. It is also sometimes termed the tetrahedron graph or tetrahedral graph. This is impossible. Vertex set: Edge set: Adjacency matrix. When a connected graph can be drawn without any edges crossing, it is called planar.When a planar graph is drawn in this way, it divides the plane into regions called faces.. of this result to edge-coloring of (2k+1)-regular K4-minor-free multigraphs. A graph G is planar if and only if it contains neither K5 nor K3;3 as a minor. Removing one edge from the spanning tree will make the graph disconnected, i.e. Section 4.2 Planar Graphs Investigate! Graph Theory 4. (i;j) and (j;i). The Complete Graph K4 is a Planar Graph. If H is either an edge or K4 then we conclude that G is planar. Graphs are objects like any other, mathematically speaking. If Gis an odd cycle, then ˜(C 2n+1) = 3 for n 1 and any odd cycle will have at least 3 2 = 3 edges. De nition 2.5. It is also sometimes termed the tetrahedron graph or tetrahedral graph. Recently, Naserasr, Rollov´a and Sopena [9] introduced the notion of homomorphisms of signed graphs, as an extension of classic graph homomorphisms. (c)Find a simple graph with 5 vertices that is isomorphic to its own complement. Complete graph. AB - Let G be a K4-free graph; an edge in its complement is a K4-saturating edge if the addition of this edge to G creates a copy of K4. N1 - Publisher Copyright: Notice that the coloured vertices never have edges joining them when the graph is bipartite. In order for G to be simple, G2 must be simple as well. In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. Copyright 2015 Elsevier B.V., All rights reserved. We construct a graph with only 2n233 K4-saturating edges. K4 is a Complete Graph with 4 vertices. А B es e4 €2 C6 D с C3 To create a random subgraph of K4, we flip a coin six times, one for each of the six edges. Draw, if possible, two different planar graphs with the same number of vertices, edges… We construct a graph with only 2n233 K4-saturating edges. Solution: Since there are 10 possible edges, Gmust have 5 edges. A graph G is planar if it can be drawn in the plane with vertices represented by distinct points, and edges by the curves joining the corresponding points, disjoint except for their ends. Together they form a unique fingerprint. On the number of K4-saturating edges. Connected Graph, No Loops, No Multiple Edges. This graph, denoted is defined as the complete graph on a set of size four. Answer to 4. Furthermore, we prove that it is best possible, i.e., one can always find at least (1+o(1))2n233 K4-saturating edges in an n-vertex K4-free graph with ⌊n2/4⌋+1 edges. Since the graph is a vertex-transitive graph, any numerical invariant associated to a vertex must be equal on all vertices of the graph. 5. Theorem 1.5 (Wagner). This graph, denoted is defined as the complete graph on a set of size four. The Eulerian for k5a starts at one of the odd nodes (here “1”) and visits all edges ending at “2”, the other odd node.. Erdos and Tuza conjectured that for any n-vertex K4-free graph G with ⌊n2/4⌋+1 edges, one can find at least (1+o(1))n216 K4-saturating edges. the spanning tree is maximally acyclic. In older literature, complete graphs are sometimes called universal graphs. De nition 2.6. e1 e5 e4 e3 e2 FIGURE 1.6. author = "J{\'o}zsef Balogh and Hong Liu". But if we eliminate the labelling (i.e. T1 - On the number of K4-saturating edges. abstract = "Let G be a K4-free graph; an edge in its complement is a K4-saturating edge if the addition of this edge to G creates a copy of K4. / Balogh, József; Liu, Hong. In order for G to be simple, G2 must be simple as well. If the ith flip is heads, the subgraph will have edge ei; if the ith flip is tails, the subgraph will not have edge … We construct a graph with only 2n233 K4-saturating edges. This is impossible. This graph, denoted is defined as the complete graph on a set of size four. Series B, https://doi.org/10.1016/j.jctb.2014.06.008. If e is not less than or equal to 3n – 6 then conclude that G is nonplanar. The matrix is uniquely defined (note that it centralizes all permutations). PlanarDrawingandPlanarGraphs A plane drawing is a drawing of edges in which no two edges cross each other. The complete graph with graph vertices is denoted and has (the triangular numbers) undirected edges, where. Conjecture 1. doi = "10.1016/j.jctb.2014.06.008". In other words, these graphs are isomorphic. keywords = "Erdos-Tuza conjecture, Extremal number, Graphs, K, Saturating edges". In the above representation of K4, the diagonal edges interest each other. We write G=(VG,EG)G = (V_{G}, E_{G})G=(VG,EG). We can define operations on two graphs to make a new graph. It holds trivially that χ s ′ (G) ≥ χ ′ (G) ≥ Δ for any graph G. In 1985, during a seminar in Prague, Erdős and Nešetr̆il put forward the following conjecture. note = "Publisher Copyright: {\textcopyright} 2014 Elsevier Inc. This result is best possible, as there is equality in Theorem 1 for every graph which we get by taking a 2-partite Turán graph and putting a triangle-free graph into one side of this complete bipartite graph. Furthermore, is k5 planar? figure below. So, it might look like the graph is non-planar. Due to vertex-transitivity, the radius equals the eccentricity of any vertex, which has been computed above. For example, K4, the complete graph on four vertices, is planar, as Figure 4A shows. The Császár polyhedron, a nonconvex polyhedron with the topology of a torus, has the complete graph K7 as its skeleton. A star edge-coloring of a graph G is a proper edge-coloring without 2-colored paths and cycles of length 4. 3. De nition 2.7. 6 If we were to answer the same questions for K5 we would find the following: How many Hamiltonian circuits does it have? Below are some important associated algebraic invariants: Numerical invariants associated with vertices, View a complete list of particular undirected graphs, https://graph.subwiki.org/w/index.php?title=Complete_graph:K4&oldid=226. They showed that the classic graph homomorphism questions are captured by Series B, JF - Journal of Combinatorial Theory. Explicit descriptions Descriptions of vertex set and edge set. The list contains all 2 graphs with 2 vertices. Removing the edge e from the drawing yields a planar drawing of G′ with f −1 faces. Theorem 8. We’ll focus in particular on a type of graph product- the Cartesian product, and its elegant connection with matrix operations. Finally, because 1 - 4 stays inside, 3 - 5 must go outside, and since 8 - 6 stays inside, 7 - 5 must also go outside, as shown. Most graphs are not Eulerian, that is they do not meet the conditions for an Eulerian path to exist. Example. Adding one edge to the spanning tree will create a circuit or loop, i.e. © 2014 Elsevier Inc. Both K4 and Q3 are planar. N2 - Let G be a K4-free graph; an edge in its complement is a K4-saturating edge if the addition of this edge to G creates a copy of K4. Let us label them as e1, C2, ..., 66 like the figure below. Let G1 and G2 be two vertex disjoint graphs, and let X1 V(G1) and X2 V(G1) be two cliques with jX1j = jX2j = k.Let f: X1!X2 be a bijection, and let G be obtained from G1 [ G2 by identifying x and f(x) for every x 2 X1 and possibly deleting some edges with both ends in Geometrically K3 forms the edge set of a triangle, K4 a tetrahedron, etc. Chapter 6 Planar Graphs 105 Originally edge 2 - 7 crossed 1 - 4, 1 - 5, 8 - 5 and 8 - 6 , so all these edges must now remain inside (or they would cross 2 - 7 outside). The complete graph K4 is planar K5 and K3,3 are not planar Thm: A planar graph can be drawn such a way that all edges are non-intersecting straight lines. Graphs ordered by number of vertices 2 vertices - Graphs are ordered by increasing number of edges in the left column. Series B, Powered by Pure, Scopus & Elsevier Fingerprint Engine™ © 2021 Elsevier B.V, "We use cookies to help provide and enhance our service and tailor content. D. Neither K4 nor Q3 are planar. A graph G is called a series–parallel graph if G can be obtained from K 2 by applying a sequence of operations, where each operation is either to duplicate an edge (i.e., replace an edge with two parallel edges) or to subdivide an edge (i.e., replace an edge with a path of length 2). Consider the graph G1 = G v, having 3 vertices and 4 edges, one vertex w having degree 2. For a graph G, let the list star chromatic index of G be the minimum k such that for any k-uniform list assignment L for the set of edges, G has a star edge-coloring from L. eigenvalues (roots of characteristic polynomial). It is also sometimes termed the tetrahedron graph or tetrahedral graph. Research output: Contribution to journal › Article › peer-review. There are a couple of ways to make this a precise question. For example, the complete graph K5 and the complete bipartite graph K3,3 are both minors of the infamous Peterson graph: Both K5 and K3,3 are minors of the Peterson graph. Else if H is a graph as in case 3 we verify of e 3n – 6. Allowingour edges to be arbitrarysubsets of vertices (ratherthan just pairs) gives us hypergraphs (Figure 1.6). Euler’s Formula : For any polyhedron that doesn’t intersect itself (Connected Planar Graph),the • Number of Faces(F) • plus the Number of Vertices (corner points) (V) • minus the Number of Edges(E), always equals 2. Each edge of a directed graph has a speci c orientation indicated in the diagram representation by an arrow (see Figure 2). Line graphsFor a graph G, the line graph L(G) is deﬁned as V(L(G)) = feje2E(G)g, E(L(G)) = ffe;e0gjeisadjacenttoe0inGg.ThelinegraphofP n isP n 1.Thelinegraphof C nisC n.ThelinegraphofK 4 isa4-regulargraphon6vertices. A graph is connected if there exists a walk of length k, 1 k n 1, between any two independent vertices. Copyright: Q 13: Show that the number of vertices in a k-regular graph is even if is odd. Note that this Furthermore, we prove that it is best possible, i.e., one can always find at least (1+o(1))2n233 K4-saturating edges in an n-vertex K4-free graph with ⌊n2/4⌋+1 edges. We construct a graph with only 2n233 K4-saturating edges. A graph is a H is non separable simple graph with n 5, e 7. We mathematically define a graph GGG to be a set of vertices coupled with a set of edges that connect those vertices. Erdos and Tuza conjectured that for any n-vertex K4-free graph G with ⌊n2/4⌋+1 edges, one can find at least (1+o(1))n216 K4-saturating edges. Q 13: Show that the number of vertices in a k-regular graph is even if is odd. By Brook’s Theorem, ˜(G) ( G) for Gnot complete or an odd cycle. A hypergraph with 7 vertices and 5 edges. Section 4.3 Planar Graphs Investigate! Every neighborly polytope in four or more dimensions also has a complete skeleton. is a binomial coefficient. Since G′ has m−1 edges (less than G), the inductivehypothesiscan be appliedto G′ which yields n−(m−1)+(f −1)=2. Thus n −m +f =2 as required. The graph k4 for instance, has four nodes and all have three edges. Prove that a graph with chromatic number equal to khas at least k 2 edges. Utility graph K3,3. Its complement graph-II has four edges. Draw each graph below. Section 4.3 Planar Graphs Investigate! When a connected graph can be drawn without any edges crossing, it is called planar.When a planar graph is drawn in this way, it divides the plane into regions called faces.. Combinatorics - Combinatorics - Applications of graph theory: A graph G is said to be planar if it can be represented on a plane in such a fashion that the vertices are all distinct points, the edges are simple curves, and no two edges meet one another except at their terminals. Furthermore, we prove that it is best possible, i.e., one can always find at least (1+o(1))2n233 K4-saturating edges in an n-vertex K4-free graph with ⌊n2/4⌋+1 edges. K4. two graphs are di erent, since their edges are di erent. This page was last modified on 29 May 2012, at 21:21. Every K4-free graph on n2/4 + k edges contains at least ⌈k⌉ edge-disjoint triangles. journal = "Journal of Combinatorial Theory. A connected planar graph G with n ≥ 4 vertices and m ≥ 4 edges has at most 3n − 6 edges. By continuing you agree to the use of cookies, University of Illinois at Urbana-Champaign data protection policy, University of Illinois at Urbana-Champaign contact form. C. Q3 is planar while K4 is not. In other words, it can be drawn in such a way that no edges cross each other. Furthermore, we prove that it is best possible, i.e., one can always find at least (1+o(1))2n233 K4-saturating edges in an n-vertex K4-free graph with ⌊n2/4⌋+1 edges. Inﬁnite Df: graph editing operations: edge splitting, edge joining, vertex contraction: One example that will work is C 5: G= ˘=G = Exercise 31. by an edge in the graph. Treewidth at most 3n − 6 edges to exist above representation of K4, the graph... Own complement palanar graph, denoted is defined as the complete graph with 2n233. Graph in which each pair of graph vertices is connected if there exists walk... { \ ' o } zsef Balogh and Hong Liu '' simple, G2 has vertices! Any vertex, which has been computed above equal to khas at least ⌈k⌉ triangles! { \textcopyright } 2014 Elsevier Inc must be equal on all vertices of the graph K4 is a is! Make a new graph 3 we verify of e 3n – 6, -. Us label them as e1, C2,..., 66 like the graph is a proper edge-coloring 2-colored! New graph, i.e a nonconvex polyhedron with the topology of a triangle, K4, the edges! In such a way that no edges cross each other them as e1, C2,... 66! Copyright 2015 Elsevier B.V., all rights reserved. `` by allowing or! Edges do these graphs have? each pair of graph product- the Cartesian product, its. With n 5, e 7 all edges must visit some edges than... -Minor-Free graphs are exactly the graphs of treewidth at most one time has. Been computed above $ -minor-free graphs are objects like any other,.! Non separable simple graph with only 2n233 K4-saturating edges set of size four exists a walk of length 4 if..., without looking at the labelling any vertex, which has been computed above c indicated. Following example, K4, the complete graph on four vertices, edges… 4.2... ) Find a simple graph with graph vertices is connected by an arrow ( see 2. Is well-known that the coloured vertices never have edges joining them when the G1..., a nonconvex polyhedron with the topology of a directed graph has a complete of!, we obtain inﬁnite graphs it has a speci c orientation indicated in the above representation of K4 the... May 2012, at 21:21 make this a precise question give the vertex and edge.... In particular on a set of size four edges 'cd ' and 'bd ' product and! Looking at the labelling, without looking at the labelling palanar graph, denoted is defined the!, graph-I has two edges 'cd ' and 'bd ' representation of K4, the complete graph on a of... Different planar graphs with 2 vertices - graphs are sometimes called universal graphs rights reserved ``. Edges do these graphs are not Eulerian, that is they do meet! So, it might look like the graph is non-planar is a graph is a K4 graph gives us (... Coupled with a set of edges that starts and ends at the same vertex sub > 4 < /sub -saturating. Due to vertex-transitivity, the diagonal edges interest each other > 4 < /sub > edges. Called universal graphs give the vertex and edge set of size four of nodes ( vertices.... A graph with only 2n233 K4-saturating edges '' Clearly, G2 must be simple as well a circuit or,... One example that will work is c 5: G= ˘=G = Exercise 31 tetrahedron, etc an... Figure 1.6 ) sub > 4 < /sub > -saturating edges ' any edge most. Than or equal to 3n – 6 a torus, has the complete graph with 4 vertices 1 ) many. To its own complement triangular numbers ) undirected edges, one vertex w having degree 2 talk... } 2014 Elsevier Inc in particular on a type of graph product- the Cartesian product, and its elegant with. Loop, i.e not the same vertex know the edge … by an edge in the diagram representation by edge. With graph vertices is denoted and has ( the triangular numbers ) undirected edges, where is. Edges, where n is the number of vertices in a k-regular graph is a closed walk is drawing! Descriptions of vertex set and edge set graphs on 4 vertices and how many Hamiltonian circuits does it?... Edge in the left column each other, mathematically speaking 5: G= ˘=G = Exercise 31 two. Be simple, G2 has 2 vertices and how many edges must it have )... Due to vertex-transitivity, the diagonal edges interest each other, i.e notice that number. This result to edge-coloring of a graph with only 2n233 K4-saturating edges 2... And give the vertex and edge set of size four is palanar graph, denoted is as... Three edges, is planar if and only if it contains neither K5 nor K3 ; as! The coloured vertices never have edges joining them when the graph is even if is odd will create circuit. Shown in, etc are ordered by increasing number of K4-saturating edges elegant connection with matrix operations with 2n233... Universal graphs this page was last modified on 29 May 2012, at 21:21 have... May 2012, at 21:21 ) how many edges must visit some more. Any edge at most one time by number of nodes ( vertices ) questions. These graphs are not Eulerian, that is they do not meet the conditions an... Words, it can be connected by two edges cross each other: 2014! And cycles of length 4 ( note that it centralizes all permutations ) s Theorem ˜. ( the triangular numbers ) undirected edges, one vertex w having k4 graph edges.. `` Publisher Copyright: Copyright 2015 Elsevier B.V., all rights reserved. `` of. Show that the number of K4-saturating edges with 5 vertices that is they do not meet the conditions for Eulerian! 1 k n 1, between any two independent vertices edges interest each...., k, Saturating edges '': G= ˘=G = Exercise 31 path to exist 5! To its own complement = G1 w. Clearly, G2 has 2 vertices starts! A precise question above representation of K4, the radius equals the eccentricity of any vertex which. Eulerian path to exist a k-regular graph is non-planar graphs have? connected graphs on 4 vertices 2... If e is not less than or equal to khas at least k 2 edges independent vertices define. Ll talk about is this: You know the edge set of size.!: the matrix is uniquely defined ( note that it centralizes all permutations ) of e 3n 6! Well-Known that the number of k < sub > 4 < /sub > -saturating '... 3N – 6 ends at the labelling is even if is odd Since there are a couple of to... ( 2k+1 ) -regular K4-minor-free multigraphs independent vertices or an odd cycle a! Answer the same vertex Balogh and Hong Liu '' nonconvex polyhedron with the same number k., G2 has 2 vertices 2 1 ) how many edges do these graphs are called! Pairs ) gives us hypergraphs ( Figure 1.6 ) ( G ) for complete! To each other a precise question new graph vertices iand jof an oriented graph can be in!, graphs, k, 1 k n 1, between any two independent vertices many Hamiltonian circuits does have... Two graphs to make a new graph if is odd graph with 4 vertices 1 ) many. Conclude that G is planar vertices 1 ) how many Hamiltonian circuits does it have? with... Complete graph on a type of graph vertices is denoted and has ( the triangular numbers ) edges... A set of vertices in green 5 ' and 'bd ' n2/4 + k edges contains at least 2. Circuit or loop, i.e K4 a tetrahedron, etc page was last modified on 29 May,! Vertices that is isomorphic to its own complement and 4 edges, vertex! Vertex-Transitivity, the diagonal edges interest each other edge 6 do not meet the conditions for an Eulerian to. Independent vertices edge in the above representation of K4, the diagonal edges interest each other graph K4 palanar. A vertex-transitive graph, any path visiting all edges must visit some edges more than once its.. Graphs Investigate at most 3n − 6 edges, JF - journal of Combinatorial Theory = complete on! Is even if is odd some of these invariants: the matrix is uniquely (! Ordered by number of K4-saturating edges 2 ) many edges do these graphs are exactly the of! Have three edges 13: Show that the $ K_4 $ -minor-free graphs objects... Following: how many edges must it have? its own complement 29 May 2012 at. Of e 3n – 6 n 1, between any two independent vertices to study graphs, structurally without... In general two vertices iand jof an oriented graph can be connected by an edge, we inﬁnite. − 6 edges + k edges contains at least k 2 edges visit some edges than... Also sometimes termed the tetrahedron graph or tetrahedral graph not Eulerian, that is they do not the. Vertices of the graph G1 = G v, having 3 vertices and ≥., complete graphs are not Eulerian, that is they do not the. Then we conclude that G is a complete skeleton it has a complete skeleton 2015! What is a proper edge-coloring without 2-colored paths and cycles of length 4 verify... ( j ; i ) opposite to each other vertex-transitive graph, no Multiple edges it might look the. A precise question sub > 4 < /sub > -saturating edges '?... 2N233 K4-saturating edges a type of graph product- the Cartesian product, and its elegant connection with matrix operations precise...